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|---|---|---|---|---|---|---|---|---|
Find all differentiable functions $f:(0,\infty) \to (0,\infty)$ for which there is a positive real number $a$ such that $f'(\frac{a}{x})=\frac{x}{f(x)}$ for all $x>0$. | the set of functions $f$ such that $f(x) = c x^d$ for some $c > 0$ and $d > 0$, with $d = 1$ implying $c = 1$ | Show that the functions are precisely $f(x)=cx^d$ for $c,d>0$ arbitrary except that we must take $c=1$ in case $d=1$. | open Nat Set | [] | @Eq (Set (Real β Real)) answer
(@setOf (Real β Real) fun (f_1 : Real β Real) =>
@Exists Real fun (c : Real) =>
And (@GT.gt Real Real.instLT c (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))
(@Exists Real fun (d : Real) =>
And (@GT.gt Real Real.instLT d (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))
(And
(@Eq Real d (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) β
@Eq Real c (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@Set.EqOn Real Real f_1
(fun (x : Real) =>
@HMul.hMul Real Real Real (@instHMul Real Real.instMul) c
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) x d))
(@Set.Ioi Real Real.instPreorder
(@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))))))) | Set (β β β) | [
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "β x > 0, 0 < f x",
"v": null,
"name": "hf"
},
{
"t": "DifferentiableOn β f (Ioi 0)",
"v": null,
"name": "hf'"
},
{
"t": "(β a > 0, β x > 0, deriv f (a / x) = x / f x) β f β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2005_b3",
"tags": [
"analysis"
]
} |
Find the volume of the region of points $(x,y,z)$ such that
\[
(x^2 + y^2 + z^2 + 8)^2 \leq 36(x^2 + y^2).
\] | 6 * Real.pi ^ 2 | Show that the volume is $6\pi^2$. | null | [] | @Eq Real answer
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@OfNat.ofNat Real (nat_lit 6)
(@instOfNatAtLeastTwo Real (nat_lit 6) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) Real.pi
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) | β | [
{
"t": "(MeasureTheory.volume {a : β Γ β Γ β | (a.1 ^ 2 + a.2.1 ^ 2 + a.2.2 ^ 2 + 8) ^ 2 β€ 36 * (a.1 ^ 2 + a.2.1 ^ 2)}).toReal = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2006_a1",
"tags": [
"geometry"
]
} |
Let $S=\{1,2,\dots,n\}$ for some integer $n>1$. Say a permutation $\pi$ of $S$ has a \emph{local maximum} at $k \in S$ if
\begin{enumerate}
\item[(i)] $\pi(k)>\pi(k+1)$ for $k=1$;
\item[(ii)] $\pi(k-1)<\pi(k)$ and $\pi(k)>\pi(k+1)$ for $1<k<n$;
\item[(iii)] $\pi(k-1)<\pi(k)$ for $k=n$.
\end{enumerate}
(For example, if $n=5$ and $\pi$ takes values at $1,2,3,4,5$ of $2,1,4,5,3$, then $\pi$ has a local maximum of 2 at $k=1$, and a local maximum of 5 at $k=4$.) What is the average number of local maxima of a permutation of $S$, averaging over all permutations of $S$? | (fun n : β => (n + 1) / 3) | Show that the average number of local maxima is $\frac{n+1}{3}$. | null | [] | @Eq (Nat β Real) answer fun (n_1 : Nat) =>
@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) (@Nat.cast Real Real.instNatCast n_1)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))) | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "Equiv.Perm (Fin n) β (β β β)",
"v": null,
"name": "pnat"
},
{
"t": "Equiv.Perm (Fin n) β β",
"v": null,
"name": "pcount"
},
{
"t": "n > 1",
"v": null,
"name": "ngt1"
},
{
"t": "β p : Equiv.Perm (Fin n), β k : Fin n, (pnat p) k = p k",
"v": null,
"name": "hpnat"
},
{
"t": "β p : Equiv.Perm (Fin n), pcount p = {k : Fin n | (k.1 = 0 β¨ (pnat p) (k - 1) < (pnat p) k) β§ (k = n - 1 β¨ (pnat p) k > (pnat p) (k + 1))}.encard",
"v": null,
"name": "hpcount"
},
{
"t": "(β p : Equiv.Perm (Fin n), pcount p) / {p : Equiv.Perm (Fin n) | true}.ncard = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2006_a4",
"tags": [
"algebra"
]
} |
Let $n$ be a positive odd integer and let $\theta$ be a real number such that $\theta/\pi$ is irrational. Set $a_k=\tan(\theta+k\pi/n)$, $k=1,2,\dots,n$. Prove that $\frac{a_1+a_2+\cdots+a_n}{a_1a_2 \cdots a_n}$ is an integer, and determine its value. | (fun n : β => if (n β‘ 1 [MOD 4]) then n else -n) | Show that $\frac{a_1+\cdots+a_n}{a_1 \cdots a_n}=\begin{cases} n & n \equiv 1 \pmod{4} \\ -n & n \equiv 3 \pmod{4}. \end{cases}$ | null | [] | @Eq (Nat β Int) answer fun (n_1 : Nat) =>
@ite Int
(Nat.ModEq (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) n_1
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@Nat.instDecidableModEq (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) n_1
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@Nat.cast Int instNatCastInt n_1) (@Neg.neg Int Int.instNegInt (@Nat.cast Int instNatCastInt n_1)) | β β β€ | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "β",
"v": null,
"name": "theta"
},
{
"t": "Set.Icc 1 n β β",
"v": null,
"name": "a"
},
{
"t": "Odd n",
"v": null,
"name": "nodd"
},
{
"t": "Irrational (theta / Real.pi)",
"v": null,
"name": "thetairr"
},
{
"t": "β k : Set.Icc 1 n, a k = Real.tan (theta + (k * Real.pi) / n)",
"v": null,
"name": "ha"
},
{
"t": "(β k : Set.Icc 1 n, a k) / (β k : Set.Icc 1 n, a k) = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2006_a5",
"tags": [
"algebra"
]
} |
Show that the curve $x^3 + 3xy + y^3 = 1$ contains only one set of three distinct points, $A$, $B$, and $C$, which are vertices of an equilateral triangle, and find its area. | 3 * Real.sqrt 3 / 2 | Prove that the triangle has area $\frac{3 \sqrt 3}{2}$. | null | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(Real.sqrt
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) | β | [
{
"t": "Set (β Γ β)",
"v": null,
"name": "curve"
},
{
"t": "curve = {c | c.1 ^ 3 + 3 * c.1 * c.2 + c.2 ^ 3 = 1}",
"v": null,
"name": "hcurve"
},
{
"t": "Set (β Γ β) β Prop",
"v": null,
"name": "equilateral"
},
{
"t": "β S, equilateral S β S.encard = 3 β§\n β d : β, β P β S, β Q β S, P β Q β\n Real.sqrt ((P.1 - Q.1)^2 + (P.2 - Q.2)^2) = d",
"v": null,
"name": "hequilateral"
},
{
"t": "(β! S : Set (β Γ β), S β curve β§ equilateral S) β§ (β S : Set (β Γ β), S β curve β§ equilateral S β§ (MeasureTheory.volume (convexHull β S)).toReal = answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2006_b1",
"tags": [
"geometry"
]
} |
Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A \cup B=S$, $A \cap B=\emptyset$, and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_S$ be the number of linear partitions of $S$. For each positive integer $n$, find the maximum of $L_S$ over all sets $S$ of $n$ points. | $\binom{n}{2} + 1$ | Show that the maximum is $\binom{n}{2}+1$. | null | [] | @Eq (Nat β Nat) answer fun (n_1 : Nat) =>
@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat)
(Nat.choose n_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) | β β β | [
{
"t": "Finset (Fin 2 β β) β Finset (Finset (Fin 2 β β)) β Prop",
"v": null,
"name": "IsLinearPartition"
},
{
"t": "β S AB, IsLinearPartition S AB β\n (AB.card = 2 β§ β A β AB, β B β AB,\n A β B β§ (A βͺ B = S) β§ (A β© B = β
) β§\n (β m b : β,\n (β p β A, p 1 > m * p 0 + b) β§\n (β p β B, p 1 < m * p 0 + b)))",
"v": null,
"name": "IsLinearPartition_def"
},
{
"t": "Finset (Fin 2 β β) β β",
"v": null,
"name": "L"
},
{
"t": "β S, L S = {AB | IsLinearPartition S AB}.encard",
"v": null,
"name": "hL"
},
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "0 < n",
"v": null,
"name": "npos"
},
{
"t": "IsGreatest {L S | (S) (hS : S.card = n)} (answer n)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2006_b3",
"tags": [
"geometry"
]
} |
Let $Z$ denote the set of points in $\mathbb{R}^n$ whose coordinates are $0$ or $1$. (Thus $Z$ has $2^n$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^n$.) Given a vector subspace $V$ of $\mathbb{R}^n$, let $Z(V)$ denote the number of members of $Z$ that lie in $V$. Let $k$ be given, $0 \leq k \leq n$. Find the maximum, over all vector subspaces $V \subseteq \mathbb{R}^n$ of dimension $k$, of the number of points in $V \cap Z$. | $2^k$ | Prove that the maximum is $2^k$. | null | [] | @Eq (Nat β Nat) answer fun (k_1 : Nat) =>
@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) k_1 | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "β",
"v": null,
"name": "k"
},
{
"t": "n > 0",
"v": null,
"name": "npos"
},
{
"t": "k β€ n",
"v": null,
"name": "hk"
},
{
"t": "Set (Fin n β β)",
"v": null,
"name": "Z"
},
{
"t": "Z = {P : Fin n β β | β j : Fin n, P j = 0 β¨ P j = 1}",
"v": null,
"name": "hZ"
},
{
"t": "IsGreatest\n {y | β V : Subspace β (Fin n β β), Module.rank β V = k β§ (Z β© V).ncard = y}\n (answer k)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2006_b4",
"tags": [
"linear_algebra"
]
} |
For each continuous function $f: [0,1] \to \mathbb{R}$, let $I(f) = \int_0^1 x^2 f(x)\,dx$ and $J(x) = \int_0^1 x \left(f(x)\right)^2\,dx$. Find the maximum value of $I(f) - J(f)$ over all such functions $f$. | $\frac{1}{16}$ | Show that the answer is \frac{1}{16}. | open Set | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@OfNat.ofNat Real (nat_lit 16)
(@instOfNatAtLeastTwo Real (nat_lit 16) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 14) (instOfNatNat (nat_lit 14))))))) | β | [
{
"t": "(β β β) β β",
"v": null,
"name": "I"
},
{
"t": "(β β β) β β",
"v": null,
"name": "J"
},
{
"t": "I = fun f β¦ β« x in (0)..1, x ^ 2 * (f x)",
"v": null,
"name": "hI"
},
{
"t": "J = fun f β¦ β« x in (0)..1, x * (f x) ^ 2",
"v": null,
"name": "hJ"
},
{
"t": "IsGreatest {y | β f : β β β, ContinuousOn f (Icc 0 1) β§ I f - J f = y} answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2006_b5",
"tags": [
"analysis",
"algebra"
]
} |
Let $k$ be an integer greater than 1. Suppose $a_0 > 0$, and define \[ a_{n+1} = a_n + \frac{1}{\sqrt[k]{a_n}} \] for $n > 0$. Evaluate \[\lim_{n \to \infty} \frac{a_n^{k+1}}{n^k}.\] | fun k => ((k+1)/k)^k | Show that the solution is $(\frac{k+1}{k})^k$. | open Set Topology Filter | [] | @Eq (Nat β Real) answer fun (k_1 : Nat) =>
@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) (@Nat.cast Real Real.instNatCast k_1)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@Nat.cast Real Real.instNatCast k_1))
k_1 | β β β | [
{
"t": "β",
"v": null,
"name": "k"
},
{
"t": "k > 1",
"v": null,
"name": "hk"
},
{
"t": "β β β",
"v": null,
"name": "a"
},
{
"t": "a 0 > 0",
"v": null,
"name": "ha0"
},
{
"t": "β n : β, a (n + 1) = a n + 1/((a n)^((1 : β)/k))",
"v": null,
"name": "ha"
},
{
"t": "Tendsto (fun n => (a n)^(k+1)/(n ^ k)) atTop (π (answer k))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2006_b6",
"tags": [
"analysis"
]
} |
Find all values of $\alpha$ for which the curves $y = \alpha*x^2 + \alpha*x + 1/24$ and $x = \alpha*y^2 + \alpha*y + 1/24$ are tangent to each other. | {2 / 3, 3 / 2, (13 + β601) / 12, (13 - β601) / 12} | Show that the solution is the set \{2/3, 3/2, (13 + \sqrt{601})/12, (13 - \sqrt{601})/12}. | null | [] | @Eq (Set Real) answer
(@Insert.insert Real (Set Real) (@Set.instInsert Real)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
(@Insert.insert Real (Set Real) (@Set.instInsert Real)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
(@Insert.insert Real (Set Real) (@Set.instInsert Real)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@OfNat.ofNat Real (nat_lit 13)
(@instOfNatAtLeastTwo Real (nat_lit 13) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 11) (instOfNatNat (nat_lit 11))))))
(Real.sqrt
(@OfNat.ofNat Real (nat_lit 601)
(@instOfNatAtLeastTwo Real (nat_lit 601) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 599) (instOfNatNat (nat_lit 599))))))))
(@OfNat.ofNat Real (nat_lit 12)
(@instOfNatAtLeastTwo Real (nat_lit 12) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 10) (instOfNatNat (nat_lit 10)))))))
(@Singleton.singleton Real (Set Real) (@Set.instSingletonSet Real)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@OfNat.ofNat Real (nat_lit 13)
(@instOfNatAtLeastTwo Real (nat_lit 13) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 11) (instOfNatNat (nat_lit 11))))))
(Real.sqrt
(@OfNat.ofNat Real (nat_lit 601)
(@instOfNatAtLeastTwo Real (nat_lit 601) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 599) (instOfNatNat (nat_lit 599))))))))
(@OfNat.ofNat Real (nat_lit 12)
(@instOfNatAtLeastTwo Real (nat_lit 12) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 10) (instOfNatNat (nat_lit 10))))))))))) | Set β | [
{
"t": "β",
"v": null,
"name": "Ξ±"
},
{
"t": "(β β β) β Prop",
"v": null,
"name": "P"
},
{
"t": "β f, P f β β x y, f x = y β§ f y = x β§ deriv f x * deriv f y = 1",
"v": null,
"name": "P_def"
},
{
"t": "Ξ± β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2007_a1",
"tags": [
"algebra",
"geometry"
]
} |
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $xy=1$ and both branches of the hyperbola $xy=-1$. (A set $S$ in the plane is called \emph{convex} if for any two points in $S$ the line segment connecting them is contained in $S$.) | 4 | Show that the minimum is $4$. | open MeasureTheory | [] | @Eq ENNReal answer
(@OfNat.ofNat ENNReal (nat_lit 4)
(@instOfNatAtLeastTwo ENNReal (nat_lit 4)
(@AddMonoidWithOne.toNatCast ENNReal
(@AddCommMonoidWithOne.toAddMonoidWithOne ENNReal instENNRealAddCommMonoidWithOne))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) | ENNReal | [
{
"t": "Set (Fin 2 β β)",
"v": null,
"name": "S"
},
{
"t": "Convex β S",
"v": null,
"name": "h_convex"
},
{
"t": "β p β S, p 0 > 0 β§ p 1 > 0 β§ p 0 * p 1 = 1",
"v": null,
"name": "h_hyperbola1"
},
{
"t": "β p β S, p 0 < 0 β§ p 1 < 0 β§ p 0 * p 1 = 1",
"v": null,
"name": "h_hyperbola2"
},
{
"t": "β p β S, p 0 < 0 β§ p 1 > 0 β§ p 0 * p 1 = -1",
"v": null,
"name": "h_hyperbola3"
},
{
"t": "β p β S, p 0 > 0 β§ p 1 < 0 β§ p 0 * p 1 = -1",
"v": null,
"name": "h_hyperbola4"
},
{
"t": "volume S = answer",
"v": null,
"name": "h_area"
}
] | {
"problem_name": "putnam_2007_a2",
"tags": [
"geometry"
]
} |
Let $k$ be a positive integer. Suppose that the integers $1, 2, 3, \dots, 3k+1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by $3$? Your answer should be in closed form, but may include factorials. | fun k β¦ (k)! * (k + 1)! / ((3 * k + 1) * (2 * k)!) | Prove that the desired probability is $\frac{k!(k+1)!}{(3k+1)(2k)!}$. | open Set Nat | [] | @Eq (Nat β Rat) answer fun (k_1 : Nat) =>
@HDiv.hDiv Rat Rat Rat (@instHDiv Rat Rat.instDiv)
(@HMul.hMul Rat Rat Rat (@instHMul Rat Rat.instMul) (@Nat.cast Rat Rat.instNatCast (Nat.factorial k_1))
(@Nat.cast Rat Rat.instNatCast
(Nat.factorial
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) k_1
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
(@HMul.hMul Rat Rat Rat (@instHMul Rat Rat.instMul)
(@HAdd.hAdd Rat Rat Rat (@instHAdd Rat Rat.instAdd)
(@HMul.hMul Rat Rat Rat (@instHMul Rat Rat.instMul) (@OfNat.ofNat Rat (nat_lit 3) (@Rat.instOfNat (nat_lit 3)))
(@Nat.cast Rat Rat.instNatCast k_1))
(@OfNat.ofNat Rat (nat_lit 1) (@Rat.instOfNat (nat_lit 1))))
(@Nat.cast Rat Rat.instNatCast
(Nat.factorial
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
k_1)))) | β β β | [
{
"t": "β",
"v": null,
"name": "k"
},
{
"t": "k > 0",
"v": null,
"name": "kpos"
},
{
"t": "Set (Fin (3 * k + 1) β β€)",
"v": null,
"name": "perms"
},
{
"t": "Set (Fin (3 * k + 1) β β€)",
"v": null,
"name": "goodperms"
},
{
"t": "goodperms = {f β perms | Β¬β j : Fin (3 * k + 1), 3 β£ β i : Fin (3 * k + 1), ite (i β€ j) (f i) 0}",
"v": null,
"name": "hgoodperms"
},
{
"t": "perms = {f : Fin (3 * k + 1) β β€ | β y β Icc 1 (3 * k + 1), β! x : Fin (3 * k + 1), f x = y}",
"v": null,
"name": "hperms"
},
{
"t": "goodperms.ncard = perms.ncard * (answer k)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2007_a3",
"tags": [
"combinatorics"
]
} |
A \emph{repunit} is a positive integer whose digits in base 10 are all ones. Find all polynomials $f$ with real coefficients such that if $n$ is a repunit, then so is $f(n)$. | the set of polynomials of the form $f(n) = \frac{1}{9} \left(10^c (9n + 1)^d - 1\right)$ where $d \in \mathbb{N}$ and $c \geq 1 - d$ | Show that the desired polynomials $f$ are those of the form
\[
f(n) = \frac{1}{9}(10^c (9n+1)^d - 1)
\]
for integers $d \geq 0$ and $c \geq 1-d$. | open Set Nat | [] | @Eq (Set (@Polynomial Real Real.semiring)) answer
(@setOf (@Polynomial Real Real.semiring) fun (f : @Polynomial Real Real.semiring) =>
@Exists Nat fun (d : Nat) =>
@Exists Int fun (c : Int) =>
And
(@GE.ge Int Int.instLEInt c
(@HSub.hSub Int Int Int (@instHSub Int Int.instSub) (@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1)))
(@Nat.cast Int instNatCastInt d)))
(β (n : Real),
@Eq Real (@Polynomial.eval Real Real.semiring n f)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@OfNat.ofNat Real (nat_lit 9)
(@instOfNatAtLeastTwo Real (nat_lit 9) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7)))))))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@HPow.hPow Real Int Real (@instHPow Real Int (@DivInvMonoid.toZPow Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 10)
(@instOfNatAtLeastTwo Real (nat_lit 10) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))))))
c)
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@OfNat.ofNat Real (nat_lit 9)
(@instOfNatAtLeastTwo Real (nat_lit 9) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7))))))
n)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
d))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))))) | Set (Polynomial β) | [
{
"t": "Set (Polynomial β)",
"v": null,
"name": "S"
},
{
"t": "β β Prop",
"v": null,
"name": "repunit"
},
{
"t": "β x, repunit x β x > 0 β§ x = floor x β§ β d β (digits 10 (floor x)), d = 1",
"v": null,
"name": "hrepunit"
},
{
"t": "β f, f β S β (β n : β, repunit n β repunit (f.eval n))",
"v": null,
"name": "hS"
},
{
"t": "S = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2007_a4",
"tags": [
"analysis",
"algebra",
"number_theory"
]
} |
Let $x_0 = 1$ and for $n \geq 0$, let $x_{n+1} = 3x_n + \lfloor x_n \sqrt{5} \rfloor$. In particular, $x_1 = 5$, $x_2 = 26$, $x_3 = 136$, $x_4 = 712$. Find a closed-form expression for $x_{2007}$. ($\lfloor a \rfloor$ means the largest integer $\leq a$.) | $\frac{2^{2006}}{\sqrt{5}} \left( \left( \frac{1 + \sqrt{5}}{2} \right)^{3997} - \left( \frac{1 + \sqrt{5}}{2} \right)^{-3997} \right)$ | Prove that $x_{2007} = \frac{2^{2006}}{\sqrt{5}}(\alpha^{3997}-\alpha^{-3997})$, where $\alpha = \frac{1+\sqrt{5}}{2}$. | open Set Nat Function | [] | @Eq Real answer
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(@OfNat.ofNat Nat (nat_lit 2006) (instOfNatNat (nat_lit 2006))))
(Real.sqrt
(@OfNat.ofNat Real (nat_lit 5)
(@instOfNatAtLeastTwo Real (nat_lit 5) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(Real.sqrt
(@OfNat.ofNat Real (nat_lit 5)
(@instOfNatAtLeastTwo Real (nat_lit 5) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
(@OfNat.ofNat Nat (nat_lit 3997) (instOfNatNat (nat_lit 3997))))
(@HPow.hPow Real Int Real (@instHPow Real Int (@DivInvMonoid.toZPow Real Real.instDivInvMonoid))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(Real.sqrt
(@OfNat.ofNat Real (nat_lit 5)
(@instOfNatAtLeastTwo Real (nat_lit 5) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
(@Neg.neg Int Int.instNegInt (@OfNat.ofNat Int (nat_lit 3997) (@instOfNat (nat_lit 3997))))))) | β | [
{
"t": "β β β",
"v": null,
"name": "x"
},
{
"t": "x 0 = 1",
"v": null,
"name": "hx0"
},
{
"t": "β n : β, x (n + 1) = 3 * (x n) + β(x n) * Real.sqrt 5β",
"v": null,
"name": "hx"
},
{
"t": "x 2007 = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2007_b3",
"tags": [
"analysis"
]
} |
Let $n$ be a positive integer. Find the number of pairs $P, Q$ of polynomials with real coefficients such that
\[
(P(X))^2 + (Q(X))^2 = X^{2n} + 1
\]
and $\deg P > \deg Q$. | fun n β¦ 2 ^ (n + 1) | Show that the number of pairs is $2^{n+1}$. | open Set Nat Function | [] | @Eq (Nat β Nat) answer fun (n_1 : Nat) =>
@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n > 0",
"v": null,
"name": "npos"
},
{
"t": "{a : (Polynomial β) Γ (Polynomial β) | a.1 ^ 2 + a.2 ^ 2 = Polynomial.X ^ (2 * n) + 1 β§ a.1.degree > a.2.degree}.ncard = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2007_b4",
"tags": [
"algebra"
]
} |
Define $f : \mathbb{R} \to \mathbb{R} by $f(x) = x$ if $x \leq e$ and $f(x) = x * f(\ln(x))$ if $x > e$. Does $\sum_{n=1}^{\infty} 1/(f(n))$ converge? | False | Show that the sum does not converge. | open Filter Topology | [] | @Eq Prop answer False | Prop | [
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "f = fun x => if x β€ Real.exp 1 then x else x * (f (Real.log x))",
"v": null,
"name": "hf"
},
{
"t": "(β r : β, Tendsto (fun N : β => β n in Finset.range N, 1/(f (n + 1))) atTop (π r)) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2008_a4",
"tags": [
"algebra"
]
} |
What is the maximum number of rational points that can lie on a circle in $\mathbb{R}^2$ whose center is not a rational point? (A \emph{rational point} is a point both of whose coordinates are rational numbers.) | 2 | Show that the maximum number is $2$. | open Filter Topology Set | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) | β | [
{
"t": "EuclideanSpace β (Fin 2) β Prop",
"v": null,
"name": "is_rational_point"
},
{
"t": "β p : EuclideanSpace β (Fin 2), is_rational_point p β β (a b : β), a = p 0 β§ b = p 1",
"v": null,
"name": "h_rational_point"
},
{
"t": "EuclideanSpace β (Fin 2) β β β Set (EuclideanSpace β (Fin 2))",
"v": null,
"name": "real_circle"
},
{
"t": "β (c : EuclideanSpace β (Fin 2)) (r : β), real_circle c r = {p | dist p c = r}",
"v": null,
"name": "h_real_circle"
},
{
"t": "IsGreatest {n : β | β (c : EuclideanSpace β (Fin 2)) (r : β), Β¬ is_rational_point c β§ (Set.ncard {p : EuclideanSpace β (Fin 2) | p β real_circle c r β§ is_rational_point p} = n)} answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2008_b1",
"tags": [
"geometry",
"number_theory"
]
} |
Let $F_0(x)=\ln x$. For $n \geq 0$ and $x>0$, let $F_{n+1}(x)=\int_0^x F_n(t)\,dt$. Evaluate $\lim_{n \to \infty} \frac{n!F_n(1)}{\ln n}$. | -1 | Show that the desired limit is $-1$. | open Filter Topology Set Nat | [] | @Eq Real answer (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) | β | [
{
"t": "β β β β β",
"v": null,
"name": "F"
},
{
"t": "β x : β, F 0 x = Real.log x",
"v": null,
"name": "hF0"
},
{
"t": "β n : β, β x > 0, F (n + 1) x = β« t in Set.Ioo 0 x, F n t",
"v": null,
"name": "hFn"
},
{
"t": "Tendsto (fun n : β => ((n)! * F n 1) / Real.log n) atTop (π answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2008_b2",
"tags": [
"analysis"
]
} |
What is the largest possible radius of a circle contained in a $4$-dimensional hypercube of side length $1$? | β2 / 2 | Show that the answer is $\frac{\sqrt 2}{2}$. | open Metric Filter Topology Set Nat | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(Real.sqrt
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) | β | [
{
"t": "Set (EuclideanSpace β (Fin 4))",
"v": null,
"name": "H"
},
{
"t": "H = {P : Fin 4 β β | β i : Fin 4, |P i| β€ 1 / 2}",
"v": null,
"name": "H_def"
},
{
"t": "β β Prop",
"v": null,
"name": "contains"
},
{
"t": "β r, contains r β\n βα΅ (A : AffineSubspace β (EuclideanSpace β (Fin 4))) (C β A),\n Module.finrank β A.direction = 2 β§\n sphere C r β© A β H",
"v": null,
"name": "contains_def"
},
{
"t": "IsGreatest contains answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2008_b3",
"tags": [
"geometry"
]
} |
Find all continuously differentiable functions f : \mathbb{R} \to \mathbb{R} such that for every rational number $q$, the number $f(q)$ is rational and has the same denominator as $q$. | {fun (x : β) => x + n | n : β€} βͺ {fun (x : β) => -x + n | n : β€} | Show that the solution is the set of all functions of the form n + x, n - x where n is any integer. | open Filter Topology Set Nat | [] | @Eq (Set (Real β Real)) answer
(@Union.union (Set (Real β Real)) (@Set.instUnion (Real β Real))
(@setOf (Real β Real) fun (x : Real β Real) =>
@Exists Int fun (n : Int) =>
@Eq (Real β Real)
(fun (x_1 : Real) =>
@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) x_1 (@Int.cast Real Real.instIntCast n))
x)
(@setOf (Real β Real) fun (x : Real β Real) =>
@Exists Int fun (n : Int) =>
@Eq (Real β Real)
(fun (x_1 : Real) =>
@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) (@Neg.neg Real Real.instNeg x_1)
(@Int.cast Real Real.instIntCast n))
x)) | Set (β β β) | [
{
"t": "(β β β) β β β Prop",
"v": null,
"name": "fqsat"
},
{
"t": "β f q, fqsat f q β ContDiff β 1 f β§ (β p : β, p = f q β§ p.den = q.den)",
"v": null,
"name": "hfqsat"
},
{
"t": "β f : (β β β), (β q : β, fqsat f q) β f β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2008_b5",
"tags": [
"analysis"
]
} |
Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f(P)=0$ for all points $P$ in the plane? | True | Prove that $f$ is identically $0$. | open Topology MvPolynomial Filter | [] | @Eq Prop answer True | Prop | [
{
"t": "β Γ β β β",
"v": null,
"name": "f"
},
{
"t": "β O v : β Γ β, v β (0, 0) β f (O.1, O.2) + f (O.1 + v.1, O.2 + v.2) + f (O.1 + v.1 - v.2, O.2 + v.2 + v.1) + f (O.1 - v.2, O.2 + v.1) = 0",
"v": null,
"name": "h_square"
},
{
"t": "(β P : β Γ β, f P = 0) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2009_a1",
"tags": [
"geometry",
"algebra"
]
} |
Functions $f,g,h$ are differentiable on some open interval around $0$
and satisfy the equations and initial conditions
\begin{gather*}
f' = 2f^2gh+\frac{1}{gh},\quad f(0)=1, \\
g'=fg^2h+\frac{4}{fh}, \quad g(0)=1, \\
h'=3fgh^2+\frac{1}{fg}, \quad h(0)=1.
\end{gather*}
Find an explicit formula for $f(x)$, valid in some open interval around $0$. | $2^{-1/12} \left(\frac{\sin(6x+\pi/4)}{\cos^2(6x+\pi/4)}\right)^{1/6}$ | Prove that the formula is
\[
f(x) = 2^{-1/12} \left(\frac{\sin(6x+\pi/4)}{\cos^2(6x+\pi/4)}\right)^{1/6}.
\] | open Topology MvPolynomial Filter Set | [] | @Eq (Real β Real) answer fun (x : Real) =>
@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@OfNat.ofNat Real (nat_lit 12)
(@instOfNatAtLeastTwo Real (nat_lit 12) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 10) (instOfNatNat (nat_lit 10))))))))
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(Real.sin
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@OfNat.ofNat Real (nat_lit 6)
(@instOfNatAtLeastTwo Real (nat_lit 6) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))))
x)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(Real.cos
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@OfNat.ofNat Real (nat_lit 6)
(@instOfNatAtLeastTwo Real (nat_lit 6) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))))
x)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@OfNat.ofNat Real (nat_lit 6)
(@instOfNatAtLeastTwo Real (nat_lit 6) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))))))) | β β β | [
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "β β β",
"v": null,
"name": "g"
},
{
"t": "β β β",
"v": null,
"name": "h"
},
{
"t": "β",
"v": null,
"name": "a"
},
{
"t": "β",
"v": null,
"name": "b"
},
{
"t": "0 β Ioo a b",
"v": null,
"name": "hab"
},
{
"t": "DifferentiableOn β f (Ioo a b) β§ DifferentiableOn β g (Ioo a b) β§ DifferentiableOn β h (Ioo a b)",
"v": null,
"name": "hdiff"
},
{
"t": "(β x β Ioo a b, deriv f x = 2 * (f x)^2 * (g x) * (h x) + 1 / ((g x) * (h x))) β§ f 0 = 1",
"v": null,
"name": "hf"
},
{
"t": "(β x β Ioo a b, deriv g x = (f x) * (g x)^2 * (h x) + 4 / ((f x) * (h x))) β§ g 0 = 1",
"v": null,
"name": "hg"
},
{
"t": "(β x β Ioo a b, deriv h x = 3 * (f x) * (g x) * (h x)^2 + 1 / ((f x) * (g x))) β§ h 0 = 1",
"v": null,
"name": "hh"
},
{
"t": "β c d : β, 0 β Ioo c d β§ β x β Ioo c d, f x = answer x",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2009_a2",
"tags": [
"analysis"
]
} |
Let $d_n$ be the determinant of the $n \times n$ matrix whose entries, from left to right and then from top to bottom, are $\cos 1, \cos 2, \dots, \cos n^2$. (For example,\[ d_3 = \left|\begin{matrix} \cos 1 & \cos 2 & \cos 3 \\ \cos 4 & \cos 5 & \cos 6 \\ \cos 7 & \cos 8 & \cos 9 \end{matrix} \right|. \]The argument of $\cos$ is always in radians, not degrees.) Evaluate $\lim_{n\to\infty} d_n$. | 0 | Show that the limit is 0. | open Topology MvPolynomial Filter Set | [] | @Eq Real answer (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) | β | [
{
"t": "(n : β) β Matrix (Fin n) (Fin n) β",
"v": null,
"name": "cos_matrix"
},
{
"t": "β n : β, β i j : Fin n, (cos_matrix n) i j = Real.cos (1 + n * i + j)",
"v": null,
"name": "hM"
},
{
"t": "Tendsto (fun n => (cos_matrix n).det) atTop (π answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2009_a3",
"tags": [
"linear_algebra",
"analysis"
]
} |
Let $S$ be a set of rational numbers such that
\begin{enumerate}
\item[(a)] $0 \in S$;
\item[(b)] If $x \in S$ then $x+1\in S$ and $x-1\in S$; and
\item[(c)] If $x\in S$ and $x\not\in\{0,1\}$, then $\frac{1}{x(x-1)}\in S$.
\end{enumerate}
Must $S$ contain all rational numbers? | False | Prove that $S$ need not contain all rationals. | open Topology MvPolynomial Filter Set | [] | @Eq Prop answer False | Prop | [
{
"t": "Set β",
"v": null,
"name": "S"
},
{
"t": "0 β S",
"v": null,
"name": "h0"
},
{
"t": "β x β S, x + 1 β S β§ x - 1 β S",
"v": null,
"name": "h1"
},
{
"t": "β x β S, x β ({0, 1} : Set β) β 1 / (x * (x - 1)) β S",
"v": null,
"name": "h2"
},
{
"t": "(β r : β, r β S) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2009_a4",
"tags": [
"number_theory"
]
} |
Is there a finite abelian group $G$ such that the product of the orders of all its elements is 2^{2009}? | False | Show that the answer is no such finite abelian group exists. | open Topology MvPolynomial Filter Set | [] | @Eq Prop answer False | Prop | [
{
"t": "answer β (β (G : Type*) (_ : CommGroup G) (_ : Fintype G), β g : G, orderOf g = 2^2009)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2009_a5",
"tags": [
"abstract_algebra"
]
} |
A game involves jumping to the right on the real number line. If $a$ and $b$ are real numbers and $b > a$, the cost of jumping from $a$ to $b$ is $b^3-ab^2$. For what real numbers $c$ can one travel from $0$ to $1$ in a finite number of jumps with total cost exactly $c$? | the interval (1/3, 1] | Prove that the possible costs are $1/3 < c \leq 1.$ | open Topology MvPolynomial Filter Set | [] | @Eq (Set Real) answer
(@Set.Ioc Real Real.instPreorder
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) | Set β | [
{
"t": "{c : β | β s : β β β, s 0 = 0 β§ StrictMono s β§ (β n : β, s n = 1 β§ ((β i in Finset.range n, ((s (i + 1)) ^ 3 - (s i) * (s (i + 1)) ^ 2)) = c))} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2009_b2",
"tags": [
"analysis",
"algebra"
]
} |
Call a subset $S$ of $\{1, 2, \dots, n\}$ \emph{mediocre} if it has the following property: Whenever $a$ and $b$ are elements of $S$ whose average is an integer, that average is also an element of $S$. Let $A(n)$ be the number of mediocre subsets of $\{1,2,\dots,n\}$. [For instance, every subset of $\{1,2,3\}$ except $\{1,3\}$ is mediocre, so $A(3) = 7$.] Find all positive integers $n$ such that $A(n+2) - 2A(n+1) + A(n) = 1$. | {n : β€ | β k β₯ 1, n = 2 ^ k - 1} | Show that the answer is $n = 2^k - 1$ for some integer $k$. | open Topology MvPolynomial Filter Set | [] | @Eq (Set Int) answer
(@setOf Int fun (n : Int) =>
@Exists Nat fun (k : Nat) =>
And (@GE.ge Nat instLENat k (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@Eq Int n
(@HSub.hSub Int Int Int (@instHSub Int Int.instSub)
(@HPow.hPow Int Nat Int (@instHPow Int Nat (@Monoid.toNatPow Int Int.instMonoid))
(@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2))) k)
(@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1)))))) | Set β€ | [
{
"t": "β€ β Set β€ β Prop",
"v": null,
"name": "mediocre"
},
{
"t": "β n S, mediocre n S β (S β Icc 1 n) β§ β a β S, β b β S, 2 β£ a + b β (a + b) / 2 β S",
"v": null,
"name": "hmediocre"
},
{
"t": "β€ β β€",
"v": null,
"name": "A"
},
{
"t": "A = fun n β¦ ({S : Set β€ | mediocre n S}.ncard : β€)",
"v": null,
"name": "hA"
},
{
"t": "{n : β€ | n > 0 β§ A (n + 2) - 2 * A (n + 1) + A n = 1} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2009_b3",
"tags": [
"number_theory"
]
} |
Say that a polynomial with real coefficients in two variables, $x,y$, is \emph{balanced} if the average value of the polynomial on each circle centered at the origin is $0$. The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$. Find the dimension of $V$. | 2020050 | Prove that the dimension of $V$ is $2020050$. | open intervalIntegral MvPolynomial Real | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 2020050) (instOfNatNat (nat_lit 2020050))) | β | [
{
"t": "MvPolynomial (Fin 2) β β Prop",
"v": null,
"name": "IsBalanced"
},
{
"t": "β P, IsBalanced P β β r > 0,\n (β« t in (0 : β)..(2 * Ο), eval ![r * cos t, r * sin t] P) / (2 * Ο * r) = 0",
"v": null,
"name": "IsBalanced_def"
},
{
"t": "Submodule β (MvPolynomial (Fin 2) β)",
"v": null,
"name": "V"
},
{
"t": "β P, P β V β IsBalanced P β§ P.totalDegree β€ 2009",
"v": null,
"name": "V_def"
},
{
"t": "Module.rank β V = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2009_b4",
"tags": [
"algebra",
"linear_algebra"
]
} |
Given a positive integer $n$, what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8$, the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is \emph{at least} $3$.] | the ceiling of n/2 | Show that the largest such $k$ is $\lceil \frac{n}{2} \rceil$. | null | [] | @Eq (Nat β Nat) answer fun (n_1 : Nat) =>
@Nat.ceil Real Real.orderedSemiring (@FloorRing.toFloorSemiring Real Real.instLinearOrderedRing Real.instFloorRing)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@Nat.cast Real Real.instNatCast n_1)
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "β β Prop",
"v": null,
"name": "kboxes"
},
{
"t": "n > 0",
"v": null,
"name": "npos"
},
{
"t": "β k : β, kboxes k =\n (β boxes : Finset.Icc 1 n β Fin k, β i j : Fin k,\n β x in Finset.univ.filter (boxes Β· = i), (x : β) =\n β x in Finset.univ.filter (boxes Β· = j), (x : β))",
"v": null,
"name": "hkboxes"
},
{
"t": "IsGreatest kboxes (answer n)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2010_a1",
"tags": [
"algebra"
]
} |
Find all differentiable functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[
f'(x) = \frac{f(x+n)-f(x)}{n}
\]
for all real numbers $x$ and all positive integers $n$. | {f : β β β | β c d : β, β x : β, f x = c*x + d} | The solution consists of all functions of the form $f(x) = cx+d$ for some real numbers $c,d$. | null | [] | @Eq (Set (Real β Real)) answer
(@setOf (Real β Real) fun (f : Real β Real) =>
@Exists Real fun (c : Real) =>
@Exists Real fun (d : Real) =>
β (x : Real),
@Eq Real (f x)
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul) c x) d)) | Set (β β β) | [
{
"t": "{f : β β β | Differentiable β f β§\nβ x : β, β n : β€, n > 0 β deriv f x = (f (x + n) - f x)/n} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2010_a2",
"tags": [
"analysis"
]
} |
Is there an infinite sequence of real numbers $a_1, a_2, a_3, \dots$ such that \[ a_1^m + a_2^m + a_3^m + \cdots = m \] for every positive integer $m$? | False | Show that the solution is no such infinite sequence exists. | open Filter Topology Set | [] | @Eq Prop answer False | Prop | [
{
"t": "answer = (β a : β β β, β m : β, m > 0 β β' i : β, (a i)^m = m)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2010_b1",
"tags": [
"analysis"
]
} |
Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$? | 3 | Show that the smallest distance is $3$. | open Filter Topology Set | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) | β | [
{
"t": "EuclideanSpace β (Fin 2)",
"v": null,
"name": "A"
},
{
"t": "EuclideanSpace β (Fin 2)",
"v": null,
"name": "B"
},
{
"t": "EuclideanSpace β (Fin 2)",
"v": null,
"name": "C"
},
{
"t": "β i : Fin 2, A i = round (A i) β§ B i = round (B i) β§ C i = round (C i)",
"v": null,
"name": "hABCintcoords"
},
{
"t": "dist A B = round (dist A B) β§ dist A C = round (dist A C) β§ dist B C = round (dist B C)",
"v": null,
"name": "hABCintdists"
},
{
"t": "Β¬Collinear β {A, B, C}",
"v": null,
"name": "hABCall"
},
{
"t": "answer = dist A B",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2010_b2",
"tags": [
"geometry"
]
} |
There are $2010$ boxes labeled $B_1, B_2, \dots, B_{2010}$, and $2010n$ balls have been distributed among them, for some positive integer $n$. You may redistribute the balls by a sequence of moves, each of which consists of choosing an $i$ and moving \emph{exactly} $i$ balls from box $B_i$ into any one other box. For which values of $n$ is it possible to reach the distribution with exactly $n$ balls in each box, regardless of the initial distribution of balls? | {1005} | Prove that it is possible if and only if $n \geq 1005$. | open Filter Topology Set | [] | @Eq (Set Nat) answer
(@Singleton.singleton Nat (Set Nat) (@Set.instSingletonSet Nat)
(@OfNat.ofNat Nat (nat_lit 1005) (instOfNatNat (nat_lit 1005)))) | Set β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n > 0",
"v": null,
"name": "hn"
},
{
"t": "(β β Fin 2010 β β) β β β Prop",
"v": null,
"name": "trans"
},
{
"t": "β P T, trans P T β β t : β, t < T β β i j,\n i β j β§\n P t i β₯ i.1 + 1 β§ P (t + 1) i = P t i - (i.1 + 1) β§ P (t + 1) j = P t j + (i.1 + 1) β§\n β k : Fin 2010, k β i β k β j β P (t + 1) k = P t k",
"v": null,
"name": "htrans"
},
{
"t": "(β B, β i, B i = 2010 * n β βα΅ (P) (T), P 0 = B β§ trans P T β§ β i, P T i = n) β n β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2010_b3",
"tags": [
"analysis"
]
} |
Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which $p(x)q(x+1)-p(x+1)q(x)=1$. | {(p, q) : Polynomial β Γ Polynomial β | p.degree β€ 1 β§ q.degree β€ 1 β§ p.coeff 0 * q.coeff 1 - p.coeff 1 * q.coeff 0 = 1} | Show that the pairs $(p,q)$ satisfying the given equation are those of the form $p(x)=ax+b,q(x)=cx+d$ for $a,b,c,d \in \mathbb{R}$ such that $bc-ad=1$. | open Filter Topology Set | [] | @Eq (Set (Prod (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring))) answer
(@setOf (Prod (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring))
fun (x : Prod (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)) =>
_example.match_1 (fun (x_1 : Prod (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)) => Prop) x
fun (p_1 q_1 : @Polynomial Real Real.semiring) =>
And
(@LE.le (WithBot Nat) (@Preorder.toLE (WithBot Nat) (@WithBot.preorder Nat Nat.instPreorder))
(@Polynomial.degree Real Real.semiring p_1)
(@OfNat.ofNat (WithBot Nat) (nat_lit 1)
(@One.toOfNat1 (WithBot Nat) (@WithBot.one Nat (@AddMonoidWithOne.toOne Nat Nat.instAddMonoidWithOne)))))
(And
(@LE.le (WithBot Nat) (@Preorder.toLE (WithBot Nat) (@WithBot.preorder Nat Nat.instPreorder))
(@Polynomial.degree Real Real.semiring q_1)
(@OfNat.ofNat (WithBot Nat) (nat_lit 1)
(@One.toOfNat1 (WithBot Nat) (@WithBot.one Nat (@AddMonoidWithOne.toOne Nat Nat.instAddMonoidWithOne)))))
(@Eq Real
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@Polynomial.coeff Real Real.semiring p_1 (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))
(@Polynomial.coeff Real Real.semiring q_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@Polynomial.coeff Real Real.semiring p_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@Polynomial.coeff Real Real.semiring q_1 (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))) | Set (Polynomial β Γ Polynomial β) | [
{
"t": "Polynomial β",
"v": null,
"name": "p"
},
{
"t": "Polynomial β",
"v": null,
"name": "q"
},
{
"t": "(β x : β, p.eval x * q.eval (x + 1) - p.eval (x + 1) * q.eval x = 1) β (p, q) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2010_b4",
"tags": [
"algebra"
]
} |
Is there a strictly increasing function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x) = f(f(x))$ for all $x$? | False | Show that the solution is no such function exists. | open Filter Topology Set | [] | @Eq Prop answer False | Prop | [
{
"t": "(β f : β β β, StrictMono f β§ Differentiable β f β§ (β x : β, deriv f x = f (f x))) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2010_b5",
"tags": [
"analysis"
]
} |
Define a \emph{growing spiral} in the plane to be a sequence of points with integer coordinates $P_0=(0,0),P_1,\dots,P_n$ such that $n \geq 2$ and:
\begin{itemize}
\item the directed line segments $P_0P_1,P_1P_2,\dots,P_{n-1}P_n$ are in the successive coordinate directions east (for $P_0P_1$), north, west, south, east, etc.;
\item the lengths of these line segments are positive and strictly increasing.
\end{itemize}
How many of the points $(x,y)$ with integer coordinates $0 \leq x \leq 2011,0 \leq y \leq 2011$ \emph{cannot} be the last point, $P_n$ of any growing spiral? | 10053 | Show that there are $10053$ excluded points. | null | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 10053) (instOfNatNat (nat_lit 10053))) | β | [
{
"t": "List (Fin 2 β β€) β Prop",
"v": null,
"name": "IsSpiral"
},
{
"t": "β P, IsSpiral P β P.length β₯ 3 β§ P[0]! = 0 β§\n (β l : Fin (P.length - 1) β β, l > 0 β§ StrictMono l β§ (β i : Fin (P.length - 1),\n (i.1 % 4 = 0 β (P[i] 0 + l i = P[i.1 + 1]! 0 β§ P[i] 1 = P[i.1 + 1]! 1)) β§\n (i.1 % 4 = 1 β (P[i] 0 = P[i.1 + 1]! 0 β§ P[i] 1 + l i = P[i.1 + 1]! 1)) β§\n (i.1 % 4 = 2 β (P[i] 0 - l i = P[i.1 + 1]! 0 β§ P[i] 1 = P[i.1 + 1]! 1)) β§\n (i.1 % 4 = 3 β (P[i] 0 = P[i.1 + 1]! 0 β§ P[i] 1 - l i = P[i.1 + 1]! 1))))",
"v": null,
"name": "IsSpiral_def"
},
{
"t": "{p | 0 β€ p 0 β§ p 0 β€ 2011 β§ 0 β€ p 1 β§ p 1 β€ 2011 β§ Β¬β spiral, IsSpiral spiral β§ spiral.getLast! = p}.encard = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2011_a1",
"tags": [
"geometry",
"algebra"
]
} |
Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1 = b_1 = 1$ and $b_n = b_{n-1} a_n - 2$ for$n=2,3,\dots$. Assume that the sequence $(b_j)$ is bounded. Prove tha \[ S = \sum_{n=1}^\infty \frac{1}{a_1...a_n} \] converges, and evaluate $S$. | 3/2 | Show that the solution is $S = 3/2$. | open Topology Filter | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) | β | [
{
"t": "β β β",
"v": null,
"name": "a"
},
{
"t": "β β β",
"v": null,
"name": "b"
},
{
"t": "β n : β, a n > 0 β§ b n > 0",
"v": null,
"name": "habn"
},
{
"t": "a 0 = 1 β§ b 0 = 1",
"v": null,
"name": "hab1"
},
{
"t": "β n β₯ 1, b n = b (n-1) * a n - 2",
"v": null,
"name": "hb"
},
{
"t": "β B : β, β n : β, |b n| β€ B",
"v": null,
"name": "hbnd"
},
{
"t": "Tendsto (fun n => β i : Fin n, 1/(β j : Fin (i + 1), (a j))) atTop (π answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2011_a2",
"tags": [
"analysis"
]
} |
Find a real number $c$ and a positive number $L$ for which $\lim_{r \to \infty} \frac{r^c \int_0^{\pi/2} x^r\sin x\,dx}{\int_0^{\pi/2} x^r\cos x\,dx}=L$. | (-1, 2 / Ο) | Show that $(c,L)=(-1,2/\pi)$ works. | open Topology Filter | [] | @Eq (Prod Real Real) answer
(@Prod.mk Real Real (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
Real.pi)) | β Γ β | [
{
"t": "β",
"v": null,
"name": "c"
},
{
"t": "β",
"v": null,
"name": "L"
},
{
"t": "answer = (c, L)",
"v": null,
"name": "h_answer"
},
{
"t": "L > 0",
"v": null,
"name": "h_L_pos"
},
{
"t": "Tendsto (fun r : β => (r ^ c * β« x in Set.Ioo 0 (Real.pi / 2), x ^ r * Real.sin x) / (β« x in Set.Ioo 0 (Real.pi / 2), x ^ r * Real.cos x)) atTop (π L)",
"v": null,
"name": "h_limit"
}
] | {
"problem_name": "putnam_2011_a3",
"tags": [
"analysis"
]
} |
For which positive integers $n$ is there an $n \times n$ matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd? | the set of odd positive integers | Show that the answer is $n$ odd. | open Topology Filter Matrix | [] | @Eq (Set Nat) answer (@setOf Nat fun (n_1 : Nat) => @Odd Nat Nat.instSemiring n_1) | Set β | [
{
"t": "β β Prop",
"v": null,
"name": "nmat"
},
{
"t": "β n, nmat n β\n β A : Matrix (Fin n) (Fin n) β€,\n (β r, Even ((A r) β¬α΅₯ (A r))) β§\n Pairwise fun r1 r2 β¦ Odd ((A r1) β¬α΅₯ (A r2))",
"v": null,
"name": "hnmat"
},
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "0 < n",
"v": null,
"name": "npos"
},
{
"t": "nmat n β n β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2011_a4",
"tags": [
"linear_algebra"
]
} |
Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0$. Which primes appear in seven or more elements of $S$? | {2, 5} | Show that only the primes $2$ and $5$ appear seven or more times. | open Topology Filter Matrix | [] | @Eq (Set Nat) answer
(@Insert.insert Nat (Set Nat) (@Set.instInsert Nat) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Singleton.singleton Nat (Set Nat) (@Set.instSingletonSet Nat)
(@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))))) | Set β | [
{
"t": "Set (Fin 3 β β)",
"v": null,
"name": "S"
},
{
"t": "β",
"v": null,
"name": "t"
},
{
"t": "S = {s : Fin 3 β β | (s 0).Prime β§ (s 1).Prime β§ (s 2).Prime β§ β x : β, (s 0) * x ^ 2 + (s 1) * x + (s 2) = 0}",
"v": null,
"name": "hS"
},
{
"t": "(t.Prime β§ ({s β S | β i : Fin 3, s i = t}.encard β₯ 7)) β t β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2011_b2",
"tags": [
"number_theory"
]
} |
Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0$, with $g$ nonzero and continuous at $0$. If $fg$ and $f/g$ are differentiable at $0$, must $f$ be differentiable at $0$? | True | Prove that $f$ is differentiable. | open Topology Filter Matrix | [] | @Eq Prop answer True | Prop | [
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "β β β",
"v": null,
"name": "g"
},
{
"t": "g 0 β 0",
"v": null,
"name": "h_g0"
},
{
"t": "ContinuousAt g 0",
"v": null,
"name": "h_cont"
},
{
"t": "DifferentiableAt β (fun x β¦ f x * g x) 0",
"v": null,
"name": "h_diff_fg"
},
{
"t": "DifferentiableAt β (fun x β¦ f x / g x) 0",
"v": null,
"name": "h_diff_f_div_g"
},
{
"t": "DifferentiableAt β f 0 β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2011_b3",
"tags": [
"analysis"
]
} |
Let $f: [-1, 1] \to \mathbb{R}$ be a continuous function such that
\begin{itemize}
\item[(i)]
$f(x) = \frac{2-x^2}{2} f \left( \frac{x^2}{2-x^2} \right)$ for every $x$ in $[-1, 1]$,
\item[(ii)]
$f(0) = 1$, and
\item[(iii)]
$\lim_{x \to 1^-} \frac{f(x)}{\sqrt{1-x}}$ exists and is finite.
\end{itemize}
Prove that $f$ is unique, and express $f(x)$ in closed form. | $f(x) = \sqrt{1 - x^2}$ | $f(x) = \sqrt{1-x^2}$ for all $x \in [-1,1]$. | open Matrix Function | [] | @Eq (Real β Real) answer fun (x : Real) =>
Real.sqrt
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) x
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) | β β β | [
{
"t": "Set β",
"v": null,
"name": "S"
},
{
"t": "S = Set.Icc (-1 : β) 1",
"v": null,
"name": "hS"
},
{
"t": "(β β β) β Prop",
"v": null,
"name": "fsat"
},
{
"t": "fsat = fun f : β β β => ContinuousOn f S β§\n(β x β S, f x = ((2 - x^2)/2)*f (x^2/(2 - x^2))) β§ f 0 = 1 β§\n(β y : β, leftLim (fun x : β => (f x)/Real.sqrt (1 - x)) 1 = y)",
"v": null,
"name": "hfsat"
},
{
"t": "fsat answer β§ β f : β β β, fsat f β β x β S, f x = answer x",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2012_a3",
"tags": [
"analysis",
"algebra"
]
} |
Let $\FF_p$ denote the field of integers modulo a prime $p$, and let $n$ be a positive integer. Let $v$ be a fixed vector in $\FF_p^n$, let $M$ be an $n \times n$ matrix with entries of $\FF_p$, and define $G: \FF_p^n \to \FF_p^n$ by $G(x) = v + Mx$. Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x) = G(x)$ and $G^{(k+1)}(x) = G(G^{(k)}(x))$. Determine all pairs $p, n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0)$, $k=1,2,\dots,p^n$ are distinct. | {q | let β¨n, _β© := q; n = 1} βͺ {(2,2)} | Show that the solution is the pairs $(p,n)$ with $n = 1$ as well as the single pair $(2,2)$. | open Matrix Function | [] | @Eq (Set (Prod Nat Nat)) answer
(@Union.union (Set (Prod Nat Nat)) (@Set.instUnion (Prod Nat Nat))
(@setOf (Prod Nat Nat) fun (q : Prod Nat Nat) =>
_example.match_1 (fun (q_1 : Prod Nat Nat) => Prop) q fun (n_1 snd : Nat) =>
@Eq Nat n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@Singleton.singleton (Prod Nat Nat) (Set (Prod Nat Nat)) (@Set.instSingletonSet (Prod Nat Nat))
(@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) | Set (β Γ β) | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "β",
"v": null,
"name": "p"
},
{
"t": "n > 0",
"v": null,
"name": "hn"
},
{
"t": "Nat.Prime p",
"v": null,
"name": "hp"
},
{
"t": "Type*",
"v": null,
"name": "F"
},
{
"t": "Field F",
"v": null,
"name": null
},
{
"t": "Fintype F",
"v": null,
"name": null
},
{
"t": "Fintype.card F = p",
"v": null,
"name": "hK"
},
{
"t": "Matrix (Fin n) (Fin n) F β (Fin n β F) β (Fin n β F) β (Fin n β F)",
"v": null,
"name": "G"
},
{
"t": "β M v x, G M v x = v + mulVec M x",
"v": null,
"name": "hG"
},
{
"t": "(n, p) β answer β\n βα΅ (M : Matrix (Fin n) (Fin n) F) (v : (Fin n β F)),\n Β¬(β i j : Finset.range (p^n), i β j β§ (G M v)^[i + 1] 0 = (G M v)^[j + 1] 0)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2012_a5",
"tags": [
"linear_algebra"
]
} |
Let $f(x,y)$ be a continuous, real-valued function on $\mathbb{R}^2$. Suppose that, for every rectangular region $R$ of area $1$, the double integral of $f(x,y)$ over $R$ equals $0$.
Must $f(x,y)$ be identically $0$? | True | Prove that $f(x,y)$ must be identically $0$. | open Matrix Function | [] | @Eq Prop answer True | Prop | [
{
"t": "((β Γ β) β β) β Prop",
"v": null,
"name": "p"
},
{
"t": "β f, p f β\n Continuous f β§\n β x1 x2 y1 y2 : β, x2 > x1 β y2 > y1\n β (x2 - x1) * (y2 - y1) = 1 β β« x in x1..x2, β« y in y1..y2, f (x, y) = 0",
"v": null,
"name": "hp"
},
{
"t": "(β f x y, p f β f (x, y) = 0) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2012_a6",
"tags": [
"analysis"
]
} |
A round-robin tournament of $2n$ teams lasted for $2n-1$ days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the $n$ games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one winning team from each day without choosing any team more than once? | True | Show that the answer is yes. | open Matrix Function Real | [] | @Eq Prop answer True | Prop | [
{
"t": "(n : β) β (Fin (2 * n - 1) β (Fin (2 * n) β Fin (2 * n))) β Prop",
"v": null,
"name": "nmatchupsgames"
},
{
"t": "(n : β) β (Fin (2 * n - 1) β (Fin (2 * n) β Fin (2 * n))) β Prop",
"v": null,
"name": "nmatchupsall"
},
{
"t": "(n : β) β (Fin (2 * n - 1) β (Fin (2 * n) β Fin (2 * n))) β (Fin (2 * n - 1) β (Fin (2 * n) β Bool)) β Prop",
"v": null,
"name": "nmatchupswins"
},
{
"t": "(n : β) β (Fin (2 * n - 1) β (Fin (2 * n) β Fin (2 * n))) β (Fin (2 * n - 1) β (Fin (2 * n) β Bool)) β Prop",
"v": null,
"name": "nmatchupswinschoices"
},
{
"t": "β n matchups, nmatchupsall n matchups β β t1 t2, t1 β t2 β (β d, matchups d t1 = t2)",
"v": null,
"name": "hnmatchupsall"
},
{
"t": "β n matchups, nmatchupsgames n matchups β β d, β t, matchups d t β t β§ matchups d (matchups d t) = t",
"v": null,
"name": "hnmatchupsgames"
},
{
"t": "β n matchups wins, nmatchupswins n matchups wins β β d t, wins d t = !(wins d (matchups d t))",
"v": null,
"name": "hnmatchupswins"
},
{
"t": "β n matchups wins, nmatchupswinschoices n matchups wins β β choices, (β d, wins d (choices d)) β§ Function.Injective choices",
"v": null,
"name": "hnmatchupswinschoices"
},
{
"t": "(β n β₯ 1, β matchups wins,\n (nmatchupsgames n matchups β§ nmatchupsall n matchups β§ nmatchupswins n matchups wins) β\n nmatchupswinschoices n matchups wins) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2012_b3",
"tags": [
"combinatorics"
]
} |
Suppose that $a_0 = 1$ and that $a_{n+1} = a_n + e^{-a_n}$ for $n=0,1,2,\dots$. Does $a_n - \log n$
have a finite limit as $n \to \infty$? (Here $\log n = \log_e n = \ln n$.) | True | Prove that the sequence has a finite limit. | open Matrix Function Real Topology Filter | [] | @Eq Prop answer True | Prop | [
{
"t": "β β β",
"v": null,
"name": "a"
},
{
"t": "a 0 = 1",
"v": null,
"name": "ha0"
},
{
"t": "β n : β, a (n + 1) = a n + exp (-a n)",
"v": null,
"name": "han"
},
{
"t": "(β L : β, Tendsto (fun n β¦ a n - Real.log n) atTop (π L)) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2012_b4",
"tags": [
"analysis"
]
} |
For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1)=1$, $c(2n)=c(n)$, and $c(2n+1)=(-1)^nc(n)$. Find the value of $\sum_{n=1}^{2013} c(n)c(n+2)$. | -1 | Show that the desired sum is $-1$. | open Function Set | [] | @Eq Int answer (@Neg.neg Int Int.instNegInt (@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1)))) | β€ | [
{
"t": "β β β€",
"v": null,
"name": "c"
},
{
"t": "c 1 = 1",
"v": null,
"name": "hc1"
},
{
"t": "β n : β, n > 0 β c (2 * n) = c n",
"v": null,
"name": "hceven"
},
{
"t": "β n : β, n > 0 β c (2 * n + 1) = (-1) ^ n * c n",
"v": null,
"name": "hcodd"
},
{
"t": "(β n : Set.Icc 1 2013, c n * c (n.1 + 2)) = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2013_b1",
"tags": [
"algebra"
]
} |
Let $C = \bigcup_{N=1}^\infty C_N$, where $C_N$ denotes the set of those `cosine polynomials' of the form
\[
f(x) = 1 + \sum_{n=1}^N a_n \cos(2 \pi n x)
\]
for which:
\begin{enumerate}
\item[(i)]
$f(x) \geq 0$ for all real $x$, and
\item[(ii)]
$a_n = 0$ whenever $n$ is a multiple of $3$.
\end{enumerate}
Determine the maximum value of $f(0)$ as $f$ ranges through $C$, and
prove that this maximum is attained. | 3 | The maximum value of $f(0)$ is $3$. | open Function Set | [] | @Eq Real answer
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))) | β | [
{
"t": "β β Set (β β β)",
"v": null,
"name": "CN"
},
{
"t": "β N : β, CN N =\n {f : β β β |\n (β x : β, f x β₯ 0) β§ \n β a : List β, a.length = N + 1 β§ (β n : Fin (N + 1), 3 β£ (n : β) β a[n]! = 0) β§ \n β x : β, f x = 1 + β n in Finset.Icc 1 N, a[(n : β)]! * Real.cos (2*Real.pi*n*x)}",
"v": null,
"name": "hCN"
},
{
"t": "IsGreatest {f 0 | f β β N β Ici 1, CN N} answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2013_b2",
"tags": [
"algebra"
]
} |
Let $A$ be the $n \times n$ matrix whose entry in the $i$-th row and $j$-th column is $\frac{1}{\min(i,j)}$ for $1 \leq i,j \leq n$. Compute $\det(A)$. | (fun n : β => (-1) ^ (n - 1) / ((n - 1)! * (n)!)) | Show that the determinant is $\frac{(-1)^{n-1}}{(n-1)!n!}$. | open Topology Filter Nat | [] | @Eq (Nat β Real) answer fun (n_1 : Nat) =>
@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@Nat.cast Real Real.instNatCast
(Nat.factorial
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(@Nat.cast Real Real.instNatCast (Nat.factorial n_1))) | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "Matrix (Fin n) (Fin n) β",
"v": null,
"name": "A"
},
{
"t": "n > 0",
"v": null,
"name": "npos"
},
{
"t": "β i j : Fin n, A i j = 1 / min (i.1 + 1 : β) (j.1 + 1)",
"v": null,
"name": "hA"
},
{
"t": "A.det = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2014_a2",
"tags": [
"linear_algebra"
]
} |
Let \( a_0 = \frac{5}{2} \) and \( a_k = a_{k-1}^2 - 2 \) for \( k \geq 1 \). Compute \( \prod_{k=0}^{\infty} \left(1 - \frac{1}{a_k}\right) \) in closed form. | \( \frac{3}{7} \) | Show that the solution is 3/7. | open Topology Filter Nat | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(@OfNat.ofNat Real (nat_lit 7)
(@instOfNatAtLeastTwo Real (nat_lit 7) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))))))) | β | [
{
"t": "β β β",
"v": null,
"name": "a"
},
{
"t": "a 0 = 5 / 2",
"v": null,
"name": "a0"
},
{
"t": "β k β₯ 1, a k = (a (k - 1)) ^ 2 - 2",
"v": null,
"name": "ak"
},
{
"t": "Tendsto (fun n : β => β k in Finset.range n, (1 - 1 / a k)) atTop (π answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2014_a3",
"tags": [
"algebra"
]
} |
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[X\right]=1$, $E\left[X^2\right]=2$, and $E\left[X^3\right]=5$. (Here $E\left[Y\right]$ denotes the expectation of the random variable $Y$.) Determine the smallest possible value of the probability of the event $X=0$. | 1/3 | Show that the answer is $\frac{1}{3}$. | open Topology Filter Nat | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))) | β | [
{
"t": "(β β β) β Prop",
"v": null,
"name": "Xrandvar"
},
{
"t": "Xrandvar = (fun X : β β β => (β n : β, X n β Set.Icc 0 1) β§ β' n : β, X n = 1)",
"v": null,
"name": "hXrandvar"
},
{
"t": "(β β β) β (β β β) β β",
"v": null,
"name": "E"
},
{
"t": "E = (fun (X : β β β) (f : β β β) => β' n : β, f n * X n)",
"v": null,
"name": "hE"
},
{
"t": "sInf {X0 : β | β X : β β β, Xrandvar X β§ E X (fun x : β => x) = 1 β§ E X (fun x : β => x ^ 2) = 2 β§ E X (fun x : β => x ^ 3) = 5 β§ X0 = X 0} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2014_a4",
"tags": [
"probability",
"analysis"
]
} |
Let \( n \) be a positive integer. What is the largest \( k \) for which there exist \( n \times n \) matrices \( M_1, \ldots, M_k \) and \( N_1, \ldots, N_k \) with real entries such that for all \( i \) and \( j \), the matrix product \( M_i N_j \) has a zero entry somewhere on its diagonal if and only if \( i \neq j \)? | n^n | Show that the solution has the form k \<= n ^ n. | open Topology Filter Nat | [] | @Eq (Nat β Nat) answer fun (n_1 : Nat) =>
@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) n_1 n_1 | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "β β Prop",
"v": null,
"name": "kex"
},
{
"t": "n > 0",
"v": null,
"name": "npos"
},
{
"t": "β k β₯ 1, kex k = β M N : Fin k β Matrix (Fin n) (Fin n) β, β i j : Fin k, ((β p : Fin n, (M i * N j) p p = 0) β i β j)",
"v": null,
"name": "hkex"
},
{
"t": "(answer n β₯ 1 β§ kex (answer n)) β§ (β k β₯ 1, kex k β k β€ answer n)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2014_a6",
"tags": [
"linear_algebra"
]
} |
A \emph{base $10$ over-expansion} of a positive integer $N$ is an expression of the form
\[
N = d_k 10^k + d_{k-1} 10^{k-1} + \cdots + d_0 10^0
\]
with $d_k \neq 0$ and $d_i \in \{0,1,2,\dots,10\}$ for all $i$. For instance, the integer $N = 10$ has two base $10$ over-expansions: $10 = 10 \cdot 10^0$ and the usual base $10$ expansion $10 = 1 \cdot 10^1 + 0 \cdot 10^0$. Which positive integers have a unique base $10$ over-expansion? | the set of positive integers that do not contain the digit 0 in their base 10 representation | Prove that the answer is the integers with no $0$'s in their usual base $10$ expansion. | open Topology Filter Nat | [] | @Eq (Set Nat) answer
(@setOf Nat fun (n : Nat) =>
And (@GT.gt Nat instLTNat n (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))
(Not
(@Exists Nat fun (a : Nat) =>
And
(@Membership.mem Nat (List Nat) (@List.instMembership Nat)
(Nat.digits (@OfNat.ofNat Nat (nat_lit 10) (instOfNatNat (nat_lit 10))) n) a)
(@Eq Nat a (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) | Set β | [
{
"t": "β β List β β Prop",
"v": null,
"name": "overexpansion"
},
{
"t": "overexpansion = fun N d β¦ N = β i : Fin d.length, (d.get i) * 10 ^ i.1 β§ d.getLastI β 0 β§ β a β d, a β Finset.range 11",
"v": null,
"name": "hoverexpansion"
},
{
"t": "Set β",
"v": null,
"name": "S"
},
{
"t": "β N : β, N β S β N > 0 β§ β! d : List β, overexpansion N d",
"v": null,
"name": "hS"
},
{
"t": "S = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2014_b1",
"tags": [
"algebra"
]
} |
Suppose that \( f \) is a function on the interval \([1,3]\) such that \(-1 \leq f(x) \leq 1\) for all \( x \) and \( \int_{1}^{3} f(x) \, dx = 0 \). How large can \(\int_{1}^{3} \frac{f(x)}{x} \, dx \) be? | \( \log(4/3) \) | Show that the solution is log (4 / 3). | open Topology Filter Nat | [] | @Eq Real answer
(Real.log
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))) | β | [
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "β x : Set.Icc (1 : β) 3, -1 β€ f x β§ f x β€ 1",
"v": null,
"name": "h_bound"
},
{
"t": "β« x in Set.Ioo 1 3, f x = 0",
"v": null,
"name": "h_integral_zero"
},
{
"t": "β« x in Set.Ioo 1 3, (f x) / x = answer",
"v": null,
"name": "h_integral_x"
}
] | {
"problem_name": "putnam_2014_b2",
"tags": [
"analysis"
]
} |
Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n \geq 2$. Find an odd prime factor of $a_{2015}$. | 181 | Show that one possible answer is $181$. | null | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 181) (instOfNatNat (nat_lit 181))) | β | [
{
"t": "β β β€",
"v": null,
"name": "a"
},
{
"t": "a 0 = 1 β§ a 1 = 2",
"v": null,
"name": "abase"
},
{
"t": "β n β₯ 2, a n = 4 * a (n - 1) - a (n - 2)",
"v": null,
"name": "arec"
},
{
"t": "Odd answer β§ answer.Prime β§ ((answer : β€) β£ a 2015)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2015_a2",
"tags": [
"number_theory"
]
} |
Compute $\log_2 \left( \prod_{a=1}^{2015}\prod_{b=1}^{2015}(1+e^{2\pi iab/2015}) \right)$. Here $i$ is the imaginary unit (that is, $i^2=-1$). | 13725 | Show that the answer is $13725$. | null | [] | @Eq Complex answer
(@OfNat.ofNat Complex (nat_lit 13725)
(@instOfNatAtLeastTwo Complex (nat_lit 13725)
(@AddMonoidWithOne.toNatCast Complex (@AddGroupWithOne.toAddMonoidWithOne Complex Complex.addGroupWithOne))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 13723) (instOfNatNat (nat_lit 13723)))))) | β | [
{
"t": "Complex.log (β a : Fin 2015, β b : Fin 2015, (1 + Complex.exp (2 * Real.pi * Complex.I * (a.1 + 1) * (b.1 + 1) / 2015))) / Complex.log 2 = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2015_a3",
"tags": [
"number_theory",
"algebra"
]
} |
For each real number $x$, let
\[
f(x) = \sum_{n\in S_x} \frac{1}{2^n},
\]
where $S_x$ is the set of positive integers $n$ for which $\lfloor nx \rfloor$ is even. What is the largest real number $L$ such that $f(x) \geq L$ for all $x \in [0,1)$? (As usual, $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.) | 4/7 | Prove that $L = \frac{4}{7}$. | null | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
(@OfNat.ofNat Real (nat_lit 7)
(@instOfNatAtLeastTwo Real (nat_lit 7) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))))))) | β | [
{
"t": "β β Set β€",
"v": null,
"name": "S"
},
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "S = fun (x : β) β¦ {n : β€ | n > 0 β§ Even βn * xβ}",
"v": null,
"name": "hS"
},
{
"t": "f = fun (x : β) β¦ β' n : S x, 1 / 2 ^ (n : β€)",
"v": null,
"name": "hf"
},
{
"t": "IsGreatest {l : β | β x β Set.Ico 0 1, f x β₯ l} answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2015_a4",
"tags": [
"analysis"
]
} |
Given a list of the positive integers $1,2,3,4,\dots$, take the first three numbers $1,2,3$ and their sum $6$ and cross all four numbers off the list. Repeat with the three smallest remaining numbers $4,5,7$ and their sum $16$. Continue in this way, crossing off the three smallest remaining numbers and their sum, and consider the sequence of sums produced: $6,16,27,36,\dots$. Prove or disprove that there is some number in the sequence whose base $10$ representation ends with $2015$. | True | Show that such a number does exist. | null | [] | @Eq Prop answer True | Prop | [
{
"t": "β β Set β",
"v": null,
"name": "sets"
},
{
"t": "Set β β (Fin 3 β β)",
"v": null,
"name": "Smin3"
},
{
"t": "β β β",
"v": null,
"name": "sums"
},
{
"t": "sets 0 = Set.Ici 1",
"v": null,
"name": "hsets0"
},
{
"t": "β S : Set β, S.encard β₯ 3 β ((Smin3 S) 0 = sInf S β§ (Smin3 S) 1 = sInf (S \\ {(Smin3 S) 0}) β§ (Smin3 S) 2 = sInf (S \\ {(Smin3 S) 0, (Smin3 S) 1}))",
"v": null,
"name": "hmin3"
},
{
"t": "β n : β, sums n = (Smin3 (sets n)) 0 + (Smin3 (sets n)) 1 + (Smin3 (sets n)) 2",
"v": null,
"name": "hsums"
},
{
"t": "β n : β, sets (n + 1) = sets n \\ {(Smin3 (sets n)) 0, (Smin3 (sets n)) 1, (Smin3 (sets n)) 2, sums n}",
"v": null,
"name": "hsetsn"
},
{
"t": "(β n : β, List.IsPrefix [5, 1, 0, 2] (Nat.digits 10 (sums n))) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2015_b2",
"tags": [
"number_theory"
]
} |
Let $S$ be the set of all $2 \times 2$ real matrices $M=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ whose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S$. | the set of all matrices of the form Ξ± * I or Ξ² * A, where I is the identity matrix and A is a specific matrix | Show that matrices of the form $\alpha A$ or $\beta B$, where $A=\left(\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right)$, $B=\left(\begin{smallmatrix} -3 & -1 \\ 1 & 3 \end{smallmatrix}\right)$, and $\alpha,\beta \in \mathbb{R}$, are the only matrices in $S$ that satisfy the given condition. | null | [] | @Eq
(Set
(Matrix (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real))
answer
(@setOf
(Matrix (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real)
fun
(A :
Matrix (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) Real) =>
Or
(@Exists Real fun (Ξ± : Real) =>
β (i j : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))),
@Eq Real (A i j)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul) Ξ±
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))
(@Exists Real fun (Ξ² : Real) =>
And
(@Eq Real
(A
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 0)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (nat_lit 0)))
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 0)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (nat_lit 0))))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul) Ξ²
(@Neg.neg Real Real.instNeg
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))))
(And
(@Eq Real
(A
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 0)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (nat_lit 0)))
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 1)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (nat_lit 1))))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul) Ξ²
(@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))))
(And
(@Eq Real
(A
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 1)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (nat_lit 1)))
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 0)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (nat_lit 0))))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul) Ξ²
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))
(@Eq Real
(A
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 1)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (nat_lit 1)))
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 1)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) (nat_lit 1))))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul) Ξ²
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))))))) | Set (Matrix (Fin 2) (Fin 2) β) | [
{
"t": "Matrix (Fin 2) (Fin 2) β",
"v": null,
"name": "M"
},
{
"t": "Set (Matrix (Fin 2) (Fin 2) β)",
"v": null,
"name": "S"
},
{
"t": "S = {M' | (M' 0 1 - M' 0 0 = M' 1 0 - M' 0 1) β§ (M' 1 0 - M' 0 1 = M' 1 1 - M' 1 0)}",
"v": null,
"name": "hS"
},
{
"t": "M β S β§ (β k > 1, M ^ k β S) β M β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2015_b3",
"tags": [
"linear_algebra"
]
} |
Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c$. Express
\[
\sum_{(a,b,c) \in T} \frac{2^a}{3^b 5^c}
\]
as a rational number in lowest terms. | (17, 21) | The answer is $17/21$. | null | [] | @Eq (Prod Int Nat) answer
(@Prod.mk Int Nat (@OfNat.ofNat Int (nat_lit 17) (@instOfNat (nat_lit 17)))
(@OfNat.ofNat Nat (nat_lit 21) (instOfNatNat (nat_lit 21)))) | β€ Γ β | [
{
"t": "β β (β€ Γ β)",
"v": null,
"name": "quotientof"
},
{
"t": "β q : β, quotientof q = (q.num, q.den)",
"v": null,
"name": "hquotientof"
},
{
"t": "quotientof (β' t : (Fin 3 β β€), if (β n : Fin 3, t n > 0) β§ t 0 < t 1 + t 2 β§ t 1 < t 2 + t 0 β§ t 2 < t 0 + t 1\n then 2^(t 0)/(3^(t 1)*5^(t 2)) else 0) = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2015_b4",
"tags": [
"algebra"
]
} |
Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\dots,n\}$ such that
\[
|i-j| = 1 \mbox{ implies } |\pi(i) -\pi(j)| \leq 2
\]
for all $i,j$ in $\{1,2,\dots,n\}$. Show that for $n \geq 2$, the quantity
\[
P_{n+5} - P_{n+4} - P_{n+3} + P_n
\]
does not depend on $n$, and find its value. | 4 | Prove that answer is $4$. | open Function | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) | β | [
{
"t": "β β β",
"v": null,
"name": "P"
},
{
"t": "P = fun n β¦ {pi : Finset.Icc 1 n β Finset.Icc 1 n | Bijective pi β§ β i j : Finset.Icc 1 n, Nat.dist i j = 1 β Nat.dist (pi i) (pi j) β€ 2}.ncard",
"v": null,
"name": "hP"
},
{
"t": "β n : β, n β₯ 2 β (P (n + 5) : β€) - (P (n + 4) : β€) - (P (n + 3) : β€) + (P n : β€) = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2015_b5",
"tags": [
"algebra"
]
} |
For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1,\sqrt{2k})$. Evaluate $\sum_{k=1}^\infty (-1)^{k-1}\frac{A(k)}{k}$. | ΟΒ² / 16 | Show that the sum converges to $\pi^2/16$. | open Filter Topology | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) Real.pi
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(@OfNat.ofNat Real (nat_lit 16)
(@instOfNatAtLeastTwo Real (nat_lit 16) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 14) (instOfNatNat (nat_lit 14))))))) | β | [
{
"t": "β β β",
"v": null,
"name": "A"
},
{
"t": "β k > 0, A k = {j : β | Odd j β§ j β£ k β§ j < Real.sqrt (2 * k)}.encard",
"v": null,
"name": "hA"
},
{
"t": "Tendsto (fun K : β β¦ β k in Finset.Icc 1 K, (-1 : β) ^ ((k : β) - 1) * (A k / (k : β))) atTop (π answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2015_b6",
"tags": [
"analysis",
"number_theory"
]
} |
Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer \[ p^{(j)}(k) = \left. \frac{d^j}{dx^j} p(x) \right|_{x=k} \] (the $j$-th derivative of $p(x)$ at $k$) is divisible by 2016. | 8 | Show that the solution is $8$. | open Polynomial Filter Topology Real Set Nat | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))) | β | [
{
"t": "IsLeast {j : β | 0 < j β§ β P : β€[X], β k : β€, 2016 β£ (derivative^[j] P).eval k} answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2016_a1",
"tags": [
"algebra",
"number_theory"
]
} |
Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that
\[
\binom{m}{n-1} > \binom{m-1}{n}.
\]
Evaluate
\[
\lim_{n \to \infty} \frac{M(n)}{n}.
\] | (3 + β5) / 2 | Show that the answer is $\frac{3 + \sqrt{5}}{2}$. | open Polynomial Filter Topology Real Set Nat | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(Real.sqrt
(@OfNat.ofNat Real (nat_lit 5)
(@instOfNatAtLeastTwo Real (nat_lit 5) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) | β | [
{
"t": "β β β",
"v": null,
"name": "M"
},
{
"t": "β n > 0, IsGreatest {m | 0 < m β§ (m - 1).choose n < m.choose (n - 1)} (M n)",
"v": null,
"name": "hM"
},
{
"t": "Tendsto (fun n β¦ M n / (n : β)) atTop (π answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2016_a2",
"tags": [
"analysis"
]
} |
Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that
\[
f(x) + f\left( 1 - \frac{1}{x} \right) = \arctan x
\]
for all real $x \neq 0$. (As usual, $y = \arctan x$ means $-\pi/2 < y < \pi/2$ and $\tan y = x$.) Find
\[
\int_0^1 f(x)\,dx.
\] | 3Ο/8 | Prove that the answer is $\frac{3\pi}{8}$. | open Polynomial Filter Topology Real Set Nat | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
Real.pi)
(@OfNat.ofNat Real (nat_lit 8)
(@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))))) | β | [
{
"t": "β β β",
"v": null,
"name": "f"
},
{
"t": "β x : β, x β 0 β f x + f (1 - 1 / x) = Real.arctan x",
"v": null,
"name": "hf"
},
{
"t": "β« x in (0)..1, f x = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2016_a3",
"tags": [
"analysis"
]
} |
Find the smallest constant $C$ such that for every real polynomial $P(x)$ of degree $3$ that has a root in the interval $[0,1]$,
\[
\int_0^1 \left| P(x) \right|\,dx \leq C \max_{x \in [0,1]} \left| P(x) \right|.
\] | 5 / 6 | Prove that the smallest such value of $C$ is $\frac{5}{6}$. | open Polynomial Filter Topology Real Set Nat List | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 5)
(@instOfNatAtLeastTwo Real (nat_lit 5) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))
(@OfNat.ofNat Real (nat_lit 6)
(@instOfNatAtLeastTwo Real (nat_lit 6) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))))) | β | [
{
"t": "β β Prop",
"v": null,
"name": "p"
},
{
"t": "β c, p c β\n β P : Polynomial β, P.degree = 3 β\n (β x β Icc 0 1, P.eval x = 0) β\n β« x in (0)..1, |P.eval x| β€ c * (sSup {y | β x β Icc 0 1, y = |P.eval x|})",
"v": null,
"name": "hp"
},
{
"t": "IsLeast p answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2016_a6",
"tags": [
"algebra"
]
} |
Let $x_0,x_1,x_2,\dots$ be the sequence such that $x_0=1$ and for $n \geq 0$,
\[
x_{n+1} = \ln(e^{x_n} - x_n)
\]
(as usual, the function $\ln$ is the natural logarithm). Show that the infinite series
\[
x_0 + x_1 + x_2 + \cdots
\]
converges and find its sum. | exp 1 - 1 | The sum converges to $e - 1$. | open Polynomial Filter Topology Real Set Nat List | [] | @Eq Real answer
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(Real.exp (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) | β | [
{
"t": "β β β",
"v": null,
"name": "x"
},
{
"t": "x 0 = 1",
"v": null,
"name": "hx0"
},
{
"t": "β n : β, x (n + 1) = log (exp (x n) - (x n))",
"v": null,
"name": "hxn"
},
{
"t": "β' n : β, x n = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2016_b1",
"tags": [
"analysis"
]
} |
Define a positive integer $n$ to be \emph{squarish} if either $n$ is itself a perfect square or the distance from $n$ to the nearest perfect square is a perfect square. For example, $2016$ is squarish, because the nearest perfect square to $2016$ is $45^2 = 2025$ and $2025 - 2016 = 9$ is a perfect square. (Of the positive integers between $1$ and $10$, only $6$ and $7$ are not squarish.)
For a positive integer $N$, let $S(N)$ be the number of squarish integers between $1$ and $N$,
inclusive. Find positive constants $\alpha$ and $\beta$ such that
\[
\lim_{N \to \infty} \frac{S(N)}{N^\alpha} = \beta,
\]
or show that no such constants exist. | (3/4, 4/3) | Prove that the limit exists for $\alpha = \frac{3}{4}$ and equals $\beta = \frac{4}{3}$. | open Classical Polynomial Filter Topology Real Set Nat List | [] | @Eq (Prod Real Real) answer
(@Prod.mk Real Real
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))) | β Γ β | [
{
"t": "β€ β Prop",
"v": null,
"name": "squarish"
},
{
"t": "β n, squarish n β IsSquare n β¨ β w : β€, IsSquare |n - w ^ 2| β§ β v : β, |n - w ^ 2| β€ |n - v ^ 2|",
"v": null,
"name": "hsquarish"
},
{
"t": "β€ β β",
"v": null,
"name": "S"
},
{
"t": "S = fun n β¦ {i β Finset.Icc 1 n | squarish i}.card",
"v": null,
"name": "hS"
},
{
"t": "β β β β Prop",
"v": null,
"name": "p"
},
{
"t": "β Ξ± Ξ², p Ξ± Ξ² β Ξ± > 0 β§ Ξ² > 0 β§ Tendsto (fun N β¦ S N / (N : β) ^ Ξ±) atTop (π Ξ²)",
"v": null,
"name": "hp"
},
{
"t": "(β Ξ± Ξ² : β, ((Ξ±, Ξ²) = answer β p Ξ± Ξ²)) β¨ Β¬β Ξ± Ξ² : β, p Ξ± Ξ²",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2016_b2",
"tags": [
"analysis"
]
} |
Let $A$ be a $2n \times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be $0$ or $1$, each with probability $1/2$. Find the expected value of $\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A$. | (fun n : β => (2 * n)! / (4 ^ n * (n)!)) | Show that the expected value equals $\frac{(2n)!}{4^nn!}$. | open Real Set Nat | [] | @Eq (Nat β Real) answer fun (n_1 : Nat) =>
@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@Nat.cast Real Real.instNatCast
(Nat.factorial
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
n_1)))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
n_1)
(@Nat.cast Real Real.instNatCast (Nat.factorial n_1))) | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n β₯ 1",
"v": null,
"name": "npos"
},
{
"t": "Set (Matrix (Fin (2 * n)) (Fin (2 * n)) β)",
"v": null,
"name": "mats01"
},
{
"t": "mats01 = {A | β i j : Fin (2 * n), A i j = 0 β¨ A i j = 1}",
"v": null,
"name": "hmats01"
},
{
"t": "(β' A : mats01, (A.1 - (Matrix.transpose A)).det) / mats01.ncard = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2016_b4",
"tags": [
"linear_algebra",
"probability"
]
} |
Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y \in (1,\infty)$ and $x^2 \leq y \leq x^3$, then $(f(x))^2 \leq f(y) \leq (f(x))^3$. | the set of all functions of the form f(x) = x^c for some c > 0 | Show that the only such functions are the functions $f(x)=x^c$ for some $c>0$. | open Polynomial Filter Topology Real Set Nat List | [] | @Eq
(Set
(@Set.Elem Real
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) β
@Set.Elem Real
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))))
answer
(@setOf
(@Set.Elem Real
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) β
@Set.Elem Real
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))
fun
(f_1 :
@Set.Elem Real
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) β
@Set.Elem Real
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))) =>
@Exists Real fun (c : Real) =>
And (@GT.gt Real Real.instLT c (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))
(β
(x :
@Set.Elem Real
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))),
@Eq Real
(@Subtype.val Real
(fun (x_1 : Real) =>
@Membership.mem Real (Set Real) (@Set.instMembership Real)
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
x_1)
(f_1 x))
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@Subtype.val Real
(fun (x_1 : Real) =>
@Membership.mem Real (Set Real) (@Set.instMembership Real)
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
x_1)
x)
c))) | Set (Set.Ioi (1 : β) β Set.Ioi (1 : β)) | [
{
"t": "Set.Ioi (1 : β) β Set.Ioi (1 : β)",
"v": null,
"name": "f"
},
{
"t": "f β answer β\n (β x y : Set.Ioi (1 : β), ((x : β) ^ 2 β€ y β§ y β€ (x : β) ^ 3) β ((f x : β) ^ 2 β€ f y β§ f y β€ (f x : β) ^ 3))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2016_b5",
"tags": [
"algebra"
]
} |
Evaluate $\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \sum_{n=0}^\infty \frac{1}{k2^n+1}$. | 1 | Show that the desired sum equals $1$. | open Polynomial Filter Topology Real Set Nat List | [] | @Eq Real answer (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) | β | [
{
"t": "β' k : β, ((-1 : β) ^ ((k + 1 : β€) - 1) / (k + 1 : β)) * β' n : β, (1 : β) / ((k + 1) * (2 ^ n) + 1) = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2016_b6",
"tags": [
"analysis"
]
} |
Let $S$ be the smallest set of positive integers such that (a) $2$ is in $S$, (b) $n$ is in $S$ whenever $n^2$ is in $S$, and (c) $(n+5)^2$ is in $S$ whenever $n$ is in $S$. Which positive integers are not in $S$?. | {x : β€ | x > 0 β§ (x = 1 β¨ 5 β£ x)} | Show that all solutions are in the set $\{x \in \mathbb{Z}\, |\, x > 0 \land (x = 1 \lor 5 \mid x)\} | null | [] | @Eq (Set Int) answer
(@setOf Int fun (x : Int) =>
And (@GT.gt Int Int.instLTInt x (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))))
(Or (@Eq Int x (@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1))))
(@Dvd.dvd Int Int.instDvd (@OfNat.ofNat Int (nat_lit 5) (@instOfNat (nat_lit 5))) x))) | Set β€ | [
{
"t": "Set β€ β Prop",
"v": null,
"name": "IsQualifying"
},
{
"t": "β S, IsQualifying S β\n (β n β S, 0 < n) β§\n 2 β S β§\n (β n > 0, n ^ 2 β S β n β S) β§\n (β n β S, (n + 5) ^ 2 β S)",
"v": null,
"name": "IsQualifying_def"
},
{
"t": "Set β€",
"v": null,
"name": "S"
},
{
"t": "IsLeast IsQualifying S",
"v": null,
"name": "hS"
},
{
"t": "SαΆ β© {n | 0 < n} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2017_a1",
"tags": [
"number_theory"
]
} |
Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers
\[
N = a + (a+1) +(a+2) + \cdots + (a+k-1)
\]
for $k=2017$ but for no other values of $k>1$. Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of these expressions? | 16 | Prove that the smallest value of $a$ is $16$. | open Topology Filter | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 16) (instOfNatNat (nat_lit 16))) | β | [
{
"t": "β€ β β β β€",
"v": null,
"name": "S"
},
{
"t": "β€ β β β Prop",
"v": null,
"name": "p"
},
{
"t": "β€ β Prop",
"v": null,
"name": "q"
},
{
"t": "S = fun (a : β€) k β¦ β i in Finset.range k, (a + i)",
"v": null,
"name": "hS"
},
{
"t": "β N k, p N k β β a > 0, S a k = N",
"v": null,
"name": "hp"
},
{
"t": "β N, q N β p N 2017 β§ β k : β, k > 1 β k β 2017 β Β¬p N k",
"v": null,
"name": "hq"
},
{
"t": "IsLeast {a : β€ | q (S a 2017)} answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2017_b2",
"tags": [
"algebra"
]
} |
Evaluate the sum \begin{gather*} \sum_{k=0}^\infty \left( 3 \cdot \frac{\ln(4k+2)}{4k+2} - \frac{\ln(4k+3)}{4k+3} - \frac{\ln(4k+4)}{4k+4} - \frac{\ln(4k+5)}{4k+5} \right) \ = 3 \cdot \frac{\ln 2}{2} - \frac{\ln 3}{3} - \frac{\ln 4}{4} - \frac{\ln 5}{5} + 3 \cdot \frac{\ln 6}{6} - \frac{\ln 7}{7} \ - \frac{\ln 8}{8} - \frac{\ln 9}{9} + 3 \cdot \frac{\ln 10}{10} - \cdots . \end{gather*} (As usual, $\ln x$ denotes the natural logarithm of $x$.) | (log 2) ^ 2 | Prove that the sum equals $(\ln 2)^2$. | open Topology Filter Real | [] | @Eq Real answer
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(Real.log
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) | β | [
{
"t": "(β' k : β, (3 * Real.log (4 * k + 2) / (4 * k + 2) - Real.log (4 * k + 3) / (4 * k + 3) - Real.log (4 * k + 4) / (4 * k + 4) - Real.log (4 * k + 5) / (4 * k + 5)) = answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2017_b4",
"tags": [
"algebra"
]
} |
Find the number of ordered $64$-tuples $(x_0,x_1,\dots,x_{63})$ such that $x_0,x_1,\dots,x_{63}$ are distinct elements of $\{1,2,\dots,2017\}$ and
\[
x_0 + x_1 + 2x_2 + 3x_3 + \cdots + 63 x_{63}
\]
is divisible by 2017. | 2016! / 1953! - 63! * 2016 | Prove that the answer is $\frac{2016!}{1953!} - 63! \cdot 2016$ | open Topology Filter Real Function Nat | [] | @Eq Nat answer
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat)
(@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv)
(Nat.factorial (@OfNat.ofNat Nat (nat_lit 2016) (instOfNatNat (nat_lit 2016))))
(Nat.factorial (@OfNat.ofNat Nat (nat_lit 1953) (instOfNatNat (nat_lit 1953)))))
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat)
(Nat.factorial (@OfNat.ofNat Nat (nat_lit 63) (instOfNatNat (nat_lit 63))))
(@OfNat.ofNat Nat (nat_lit 2016) (instOfNatNat (nat_lit 2016))))) | β | [
{
"t": "Finset (Finset.range 64 β Finset.Icc 1 2017)",
"v": null,
"name": "S"
},
{
"t": "β x, x β S β (Injective x β§ (2017 β£ (β i : Finset.range 64, if i β€ (β¨1, by norm_numβ© : Finset.range 64) then (x i : β€) else i * (x i : β€))))",
"v": null,
"name": "hs"
},
{
"t": "S.card = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2017_b6",
"tags": [
"algebra",
"number_theory"
]
} |
Find all ordered pairs $(a,b)$ of positive integers for which $\frac{1}{a} + \frac{1}{b} = \frac{3}{2018}$. | {β¨673, 1358114β©, β¨674, 340033β©, β¨1009, 2018β©, β¨2018, 1009β©, β¨340033, 674β©, β¨1358114, 673β©} | Show that all solutions are in the set of ${(673,1358114), (674,340033), (1009,2018), (2018,1009), (340033,674), (1358114,673)}$. | null | [] | @Eq (Set (Prod Int Int)) answer
(@Insert.insert (Prod Int Int) (Set (Prod Int Int)) (@Set.instInsert (Prod Int Int))
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 673) (@instOfNat (nat_lit 673)))
(@OfNat.ofNat Int (nat_lit 1358114) (@instOfNat (nat_lit 1358114))))
(@Insert.insert (Prod Int Int) (Set (Prod Int Int)) (@Set.instInsert (Prod Int Int))
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 674) (@instOfNat (nat_lit 674)))
(@OfNat.ofNat Int (nat_lit 340033) (@instOfNat (nat_lit 340033))))
(@Insert.insert (Prod Int Int) (Set (Prod Int Int)) (@Set.instInsert (Prod Int Int))
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 1009) (@instOfNat (nat_lit 1009)))
(@OfNat.ofNat Int (nat_lit 2018) (@instOfNat (nat_lit 2018))))
(@Insert.insert (Prod Int Int) (Set (Prod Int Int)) (@Set.instInsert (Prod Int Int))
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 2018) (@instOfNat (nat_lit 2018)))
(@OfNat.ofNat Int (nat_lit 1009) (@instOfNat (nat_lit 1009))))
(@Insert.insert (Prod Int Int) (Set (Prod Int Int)) (@Set.instInsert (Prod Int Int))
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 340033) (@instOfNat (nat_lit 340033)))
(@OfNat.ofNat Int (nat_lit 674) (@instOfNat (nat_lit 674))))
(@Singleton.singleton (Prod Int Int) (Set (Prod Int Int)) (@Set.instSingletonSet (Prod Int Int))
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 1358114) (@instOfNat (nat_lit 1358114)))
(@OfNat.ofNat Int (nat_lit 673) (@instOfNat (nat_lit 673)))))))))) | Set (β€ Γ β€) | [
{
"t": "β€",
"v": null,
"name": "a"
},
{
"t": "β€",
"v": null,
"name": "b"
},
{
"t": "0 < a β§ 0 < b",
"v": null,
"name": "h"
},
{
"t": "((1 : β) / a + (1 : β) / b = (3 : β) / 2018) β (β¨a, bβ© β answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2018_a1",
"tags": [
"number_theory"
]
} |
Let \( S_1, S_2, \ldots, S_{2^n-1} \) be the nonempty subsets of \( \{1, 2, \ldots, n\} \) in some order, and let \( M \) be the \( (2^n - 1) \times (2^n - 1) \) matrix whose \((i, j)\) entry is $m_{ij} = \begin{cases} 0 & \text{if } S_i \cap S_j = \emptyset; \\ 1 & \text{otherwise}. \end{cases} $ Calculate the determinant of \( M \). | 1 if n = 1, otherwise -1 | Show that the solution is 1 if n = 1, and otherwise -1. | null | [] | @Eq (Nat β Real) answer fun (n_1 : Nat) =>
@ite Real (@Eq Nat n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(instDecidableEqNat n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "Fin (2 ^ n - 1) β Set β",
"v": null,
"name": "S"
},
{
"t": "Matrix (Fin (2 ^ n - 1)) (Fin (2 ^ n - 1)) β",
"v": null,
"name": "M"
},
{
"t": "n β₯ 1",
"v": null,
"name": "npos"
},
{
"t": "Set.range S = (Set.Icc 1 n).powerset \\ {β
}",
"v": null,
"name": "hS"
},
{
"t": "β i j, M i j = if (S i β© S j = β
) = True then 0 else 1",
"v": null,
"name": "hM"
},
{
"t": "M.det = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2018_a2",
"tags": [
"linear_algebra"
]
} |
Determine the greatest possible value of $\sum_{i=1}^{10} \cos(3x_i)$ for real numbers $x_1, x_2, \ldots, x_{10}$ satisfying $\sum_{i=1}^{10} \cos(x_i) = 0$. | 480/49 | Show that the solution is $\frac{480}{49}$ | null | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 480)
(@instOfNatAtLeastTwo Real (nat_lit 480) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 478) (instOfNatNat (nat_lit 478))))))
(@OfNat.ofNat Real (nat_lit 49)
(@instOfNatAtLeastTwo Real (nat_lit 49) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 47) (instOfNatNat (nat_lit 47))))))) | β | [
{
"t": "IsGreatest\n {β i, Real.cos (3 * x i) | (x : Fin 10 β β) (hx : β i, Real.cos (x i) = 0)}\n answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2018_a3",
"tags": [
"number_theory"
]
} |
Let $\mathcal{P}$ be the set of vectors defined by $\mathcal{P}=\left\{\left.\begin{pmatrix} a \\ b \end{pmatrix}\right| 0 \leq a \leq 2, 0 \leq b \leq 100,\text{ and }a,b \in \mathbb{Z}\right\}$. Find all $\mathbf{v} \in \mathcal{P}$ such that the set $\mathcal{P} \setminus \{\mathbf{v}\}$ obtained by omitting vector $\mathbf{v}$ from $\mathcal{P}$ can be partitioned into two sets of equal size and equal sum. | {v : Mathlib.Vector β€ 2 | β b : β€, 0 β€ b β§ b β€ 100 β§ Even b β§ v.toList = [1, b]} | Show that the answer is the collection of vectors $\begin{pmatrix} 1 \\ b \end{pmatrix}$ where $0 \leq b \leq 100$ and $b$ is even. | null | [] | @Eq (Set (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) β Int)) answer
(@setOf (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) β Int)
fun (v_1 : Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) β Int) =>
@Exists Int fun (b : Int) =>
And (@LE.le Int Int.instLEInt (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))) b)
(And (@LE.le Int Int.instLEInt b (@OfNat.ofNat Int (nat_lit 100) (@instOfNat (nat_lit 100))))
(And (@Even Int Int.instAdd b)
(@Eq (Fin (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) β Int) v_1
(@Matrix.vecCons Int (Nat.succ (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))
(@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1)))
(@Matrix.vecCons Int (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) b
(@Matrix.vecEmpty Int))))))) | Set (Fin 2 β β€) | [
{
"t": "Finset (Fin 2 β β€)",
"v": null,
"name": "P"
},
{
"t": "Finset (Fin 2 β β€)",
"v": null,
"name": "Pvdiff"
},
{
"t": "Fin 2 β β€",
"v": null,
"name": "v"
},
{
"t": "P = {v' : Fin 2 β β€ | 0 β€ v' 0 β§ v' 0 β€ 2 β§ 0 β€ v' 1 β§ v' 1 β€ 100}",
"v": null,
"name": "hP"
},
{
"t": "Pvdiff = P \\ ({v} : Finset (Fin 2 β β€))",
"v": null,
"name": "hPvdiff"
},
{
"t": "(v β P β§ (β Q R : Finset (Fin 2 β β€),\n (Q βͺ R = Pvdiff) β§ (Q β© R = β
) β§ (Q.card = R.card) β§\n (β q in Q, q 0 = β r in R, r 0) β§ (β q in Q, q 1 = β r in R, r 1)))\n β v β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2018_b1",
"tags": [
"algebra"
]
} |
Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n-1$, and $n-2$ divides $2^n - 2$. | {2^2, 2^4, 2^16, 2^256} | Show that the solution is the set $\{2^2, 2^4, 2^8, 2^16\}$. | null | [] | @Eq (Set Nat) answer
(@Insert.insert Nat (Set Nat) (@Set.instInsert Nat)
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(@Insert.insert Nat (Set Nat) (@Set.instInsert Nat)
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))
(@Insert.insert Nat (Set Nat) (@Set.instInsert Nat)
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@OfNat.ofNat Nat (nat_lit 16) (instOfNatNat (nat_lit 16))))
(@Singleton.singleton Nat (Set Nat) (@Set.instSingletonSet Nat)
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@OfNat.ofNat Nat (nat_lit 256) (instOfNatNat (nat_lit 256)))))))) | Set β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "0 < n",
"v": null,
"name": "hn"
},
{
"t": "(n < 10^100 β§ ((n : β€) β£ (2^n : β€) β§ (n - 1 : β€) β£ (2^n - 1 : β€) β§ (n - 2 : β€) β£ (2^n - 2 : β€))) β n β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2018_b3",
"tags": [
"number_theory"
]
} |
Determine all possible values of the expression
\[
A^3+B^3+C^3-3ABC
\]
where $A, B$, and $C$ are nonnegative integers. | the set of nonnegative integers not congruent to 3 or 6 modulo 9 | The answer is all nonnegative integers not congruent to $3$ or $6 \pmod{9}$. | open Topology Filter | [] | @Eq (Set Int) answer
(@setOf Int fun (n : Int) =>
And (@GE.ge Int Int.instLEInt n (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))))
(And
(Not
(Int.ModEq (@OfNat.ofNat Int (nat_lit 9) (@instOfNat (nat_lit 9))) n
(@OfNat.ofNat Int (nat_lit 3) (@instOfNat (nat_lit 3)))))
(Not
(Int.ModEq (@OfNat.ofNat Int (nat_lit 9) (@instOfNat (nat_lit 9))) n
(@OfNat.ofNat Int (nat_lit 6) (@instOfNat (nat_lit 6))))))) | Set β€ | [
{
"t": "{n : β€ | β A B C : β€, A β₯ 0 β§ B β₯ 0 β§ C β₯ 0 β§ A^3 + B^3 + C^3 - 3*A*B*C = n} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2019_a1",
"tags": [
"algebra"
]
} |
Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be
the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019} b_k z^k.
\]
Let $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\dots,z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu \geq M$ for all choices of $b_0,b_1,\dots, b_{2019}$ that satisfy
\[
1 \leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
\] | 2019^(-(1:β)/2019) | The answer is $M = 2019^{-1/2019}$. | open Topology Filter | [] | @Eq Real answer
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@OfNat.ofNat Real (nat_lit 2019)
(@instOfNatAtLeastTwo Real (nat_lit 2019) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2017) (instOfNatNat (nat_lit 2017))))))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@OfNat.ofNat Real (nat_lit 2019)
(@instOfNatAtLeastTwo Real (nat_lit 2019) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2017) (instOfNatNat (nat_lit 2017)))))))) | β | [
{
"t": "Polynomial β β Prop",
"v": null,
"name": "v"
},
{
"t": "v = fun b => b.degree = 2019 β§ 1 β€ (b.coeff 0).re β§ (b.coeff 2019).re β€ 2019 β§\n (β i : Fin 2020, (b.coeff i).im = 0) β§ (β i : Fin 2019, (b.coeff i).re < (b.coeff (i + 1)).re)",
"v": null,
"name": "hv"
},
{
"t": "Polynomial β β β",
"v": null,
"name": "ΞΌ"
},
{
"t": "ΞΌ = fun b => (Multiset.map (fun Ο : β => βΟβ) (Polynomial.roots b)).sum/2019",
"v": null,
"name": "hΞΌ"
},
{
"t": "IsGreatest {M : β | β b, v b β ΞΌ b β₯ M} answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2019_a3",
"tags": [
"algebra"
]
} |
Let $f$ be a continuous real-valued function on $\mathbb{R}^3$. Suppose that for every sphere $S$ of radius $1$, the integral of $f(x,y,z)$ over the surface of $S$ equals $0$. Must $f(x,y,z)$ be identically 0? | False | Show that the answer is no. | open MeasureTheory Metric Topology Filter | [] | @Eq Prop answer False | Prop | [
{
"t": "(EuclideanSpace β (Fin 3) β β) β Prop",
"v": null,
"name": "P"
},
{
"t": "β f, P f β β C, β« x in sphere C 1, f x βΞΌH[2] = 0",
"v": null,
"name": "P_def"
},
{
"t": "(β f, Continuous f β P f β f = 0) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2019_a4",
"tags": [
"analysis"
]
} |
Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by $q(x)=\sum_{k=1}^{p-1} a_kx^k$, where $a_k=k^{(p-1)/2}\mod{p}$. Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$. | (p - 1) / 2 | Show that the answer is $\frac{p-1}{2}$. | open Topology Filter | [] | @Eq (Nat β Nat) answer fun (p_1 : Nat) =>
@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv)
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) p_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) | β β β | [
{
"t": "β",
"v": null,
"name": "p"
},
{
"t": "Polynomial (ZMod p)",
"v": null,
"name": "q"
},
{
"t": "β β ZMod p",
"v": null,
"name": "a"
},
{
"t": "β β Polynomial (ZMod p)",
"v": null,
"name": "npoly"
},
{
"t": "Odd p",
"v": null,
"name": "podd"
},
{
"t": "p.Prime",
"v": null,
"name": "pprime"
},
{
"t": "β k : β, q.coeff k = a k",
"v": null,
"name": "hq"
},
{
"t": "a 0 = 0 β§ β k > p - 1, a k = 0",
"v": null,
"name": "ha0"
},
{
"t": "β k : Set.Icc 1 (p - 1), a k = ((k : β) ^ ((p - 1) / 2)) % p",
"v": null,
"name": "haother"
},
{
"t": "β n x, (npoly n).eval x = (x - 1) ^ n",
"v": null,
"name": "hnpoly"
},
{
"t": "IsGreatest {n | npoly n β£ q} (answer p)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2019_a5",
"tags": [
"abstract_algebra",
"number_theory",
"algebra"
]
} |
Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \geq 0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k \leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square. | 5n + 1 | Show that the answer is $5n+1$. | open Topology Filter | [] | @Eq (Nat β Nat) answer fun (n_1 : Nat) =>
@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat)
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) n_1)
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))) | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "Set (Fin 2 β β€)",
"v": null,
"name": "Pn"
},
{
"t": "(Fin 2 β β€) β EuclideanSpace β (Fin 2)",
"v": null,
"name": "pZtoR"
},
{
"t": "Finset (Fin 2 β β€) β Prop",
"v": null,
"name": "sPnsquare"
},
{
"t": "Pn = {p | (p 0 = 0 β§ p 1 = 0) β¨ (β k β€ n, (p 0) ^ 2 + (p 1) ^ 2 = 2 ^ k)}",
"v": null,
"name": "hPn"
},
{
"t": "β p i, (pZtoR p) i = p i",
"v": null,
"name": "hpZtoR"
},
{
"t": "β sPn : Finset (Fin 2 β β€), sPnsquare sPn β (sPn.card = 4 β§ β p4 : Fin 4 β (Fin 2 β β€), Set.range p4 = sPn β§ (β s > 0, β i : Fin 4, dist (pZtoR (p4 i) : EuclideanSpace β (Fin 2)) (pZtoR (p4 (i + 1))) = s) β§ (dist (pZtoR (p4 0)) (pZtoR (p4 2)) = dist (pZtoR (p4 1)) (pZtoR (p4 3))))",
"v": null,
"name": "sPnsquare_def"
},
{
"t": "{sPn : Finset (Fin 2 β β€) | (sPn : Set (Fin 2 β β€)) β Pn β§ sPnsquare sPn}.encard = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2019_b1",
"tags": [
"geometry"
]
} |
For all $n \geq 1$, let
\[
a_n = \sum_{k=1}^{n-1} \frac{\sin \left( \frac{(2k-1)\pi}{2n} \right)}{\cos^2 \left( \frac{(k-1)\pi}{2n} \right) \cos^2 \left( \frac{k\pi}{2n} \right)}.
\]
Determine
\[
\lim_{n \to \infty} \frac{a_n}{n^3}.
\] | 8/Ο^3 | The answer is $\frac{8}{\pi^3}$. | open Topology Filter Set | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 8)
(@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) Real.pi
(@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))) | β | [
{
"t": "β β β",
"v": null,
"name": "a"
},
{
"t": "a = fun n : β => β k : Icc (1 : β€) (n - 1),\nReal.sin ((2*k - 1)*Real.pi/(2*n))/((Real.cos ((k - 1)*Real.pi/(2*n))^2)*(Real.cos (k*Real.pi/(2*n))^2))",
"v": null,
"name": "ha"
},
{
"t": "Tendsto (fun n : β => (a n)/n^3) atTop (π answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2019_b2",
"tags": [
"analysis"
]
} |
Let $\mathcal{F}$ be the set of functions $f(x,y)$ that are twice continuously differentiable for $x \geq 1,y \geq 1$ and that satisfy the following two equations (where subscripts denote partial derivatives):
\begin{gather*}
xf_x+yf_y=xy\ln(xy), \\
x^2f_{xx}+y^2f_{yy}=xy.
\end{gather*}
For each $f \in \mathcal{F}$, let $m(f)=\min_{s \geq 1} (f(s+1,s+1)-f(s+1,s)-f(s,s+1)+f(s,s))$. Determine $m(f)$, and show that it is independent of the choice of $f$. | $2 \ln 2 - \frac{1}{2}$ | Show that $m(f)=2\ln 2-\frac{1}{2}$. | open Topology Filter Set Matrix | [] | @Eq Real answer
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(Real.log
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))) | β | [
{
"t": "(Fin 2 β β) β β",
"v": null,
"name": "f"
},
{
"t": "β β β β (Fin 2 β β)",
"v": null,
"name": "vec"
},
{
"t": "ContDiff β 2 f",
"v": null,
"name": "fdiff"
},
{
"t": "β x y : β, (vec x y) 0 = x β§ (vec x y 1) = y",
"v": null,
"name": "hvec"
},
{
"t": "β x β₯ 1, β y β₯ 1, x * deriv (fun x' : β => f (vec x' y)) x + y * deriv (fun y' : β => f (vec x y')) y = x * y * Real.log (x * y)",
"v": null,
"name": "feq1"
},
{
"t": "β x β₯ 1, β y β₯ 1, x ^ 2 * iteratedDeriv 2 (fun x' : β => f (vec x' y)) x + y ^ 2 * iteratedDeriv 2 (fun y' : β => f (vec x y')) y = x * y",
"v": null,
"name": "feq2"
},
{
"t": "sInf {f (vec (s + 1) (s + 1)) - f (vec (s + 1) s) - f (vec s (s + 1)) + f (vec s s) | s β₯ 1} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2019_b4",
"tags": [
"analysis"
]
} |
Let $F_m$ be the $m$th Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_m = F_{m-1} + F_{m-2}$ for all $m \geq 3$. Let $p(x)$ be the polynomial of degree $1008$ such that $p(2n + 1) = F_{2n+1}$ for $n = 0,1,2,\ldots,1008$. Find integers $j$ and $k$ such that $p(2019) = F_j - F_k$. | β¨2019, 1010β© | Show that the solution takes the form of $(j, k) = (2019, 1010)$. | open Topology Filter Set Matrix | [] | @Eq (Prod Nat Nat) answer
(@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 2019) (instOfNatNat (nat_lit 2019)))
(@OfNat.ofNat Nat (nat_lit 1010) (instOfNatNat (nat_lit 1010)))) | β Γ β | [
{
"t": "β β β€",
"v": null,
"name": "F"
},
{
"t": "Polynomial β",
"v": null,
"name": "P"
},
{
"t": "β x, x β₯ 1 β F (x + 2) = F (x + 1) + F x",
"v": null,
"name": "hF"
},
{
"t": "F 1 = 1 β§ F 2 = 1",
"v": null,
"name": "F12"
},
{
"t": "Polynomial.degree P = 1008",
"v": null,
"name": "Pdeg"
},
{
"t": "β n : β, (n β€ 1008) β P.eval (2 * n + 1 : β) = F (2 * n + 1)",
"v": null,
"name": "hp"
},
{
"t": "β j k : β, (P.eval 2019 = F j - F k) β β¨j, kβ© = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2019_b5",
"tags": [
"number_theory",
"algebra"
]
} |
Let \( \mathbb{Z}^n \) be the integer lattice in \( \mathbb{R}^n \). Two points in \( \mathbb{Z}^n \) are called neighbors if they differ by exactly 1 in one coordinate and are equal in all other coordinates. For which integers \( n \geq 1 \) does there exist a set of points \( S \subset \mathbb{Z}^n \) satisfying the following two conditions? \begin{enumerate} \item If \( p \) is in \( S \), then none of the neighbors of \( p \) is in \( S \). \item If \( p \in \mathbb{Z}^n \) is not in \( S \), then exactly one of the neighbors of \( p \) is in \( S \). \end{enumerate} | the set of all integers n β₯ 1 | Show that the statement is true for every \(n \geq 1\) | open Topology Filter Set Matrix | [] | @Eq (Set Nat) answer (@Set.Ici Nat Nat.instPreorder (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) | Set β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "(Fin n β β€) β (Fin n β β€) β Prop",
"v": null,
"name": "neighbors"
},
{
"t": "β p q, neighbors p q β (β i : Fin n, abs (p i - q i) = 1 β§ β j β i, p j = q j)",
"v": null,
"name": "neighbors_def"
},
{
"t": "(1 β€ n β§ β S : Set (Fin n β β€),\n (β p β S, β q, neighbors p q β q β S) β§ (β p β S, {q β S | neighbors p q}.encard = 1)) β n β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2019_b6",
"tags": [
"algebra"
]
} |
Find the number of positive integers $N$ satisfying: (i) $N$ is divisible by $2020$, (ii) $N$ has at most $2020$ decimal digits, (iii) The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros. | 508536 | Show that the solution is $508536$. | null | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 508536) (instOfNatNat (nat_lit 508536))) | β | [
{
"t": "β",
"v": null,
"name": "N"
},
{
"t": "2020 β£ N",
"v": null,
"name": "h_div"
},
{
"t": "Nat.log 10 N + 1 β€ 2020",
"v": null,
"name": "h_digits"
},
{
"t": "β k l, k β₯ l β§ N = β i in Finset.range (k - l + 1), 10 ^ (i + l)",
"v": null,
"name": "h_pattern"
},
{
"t": "Set.ncard {x : β | (2020 β£ x) β§ (Nat.log 10 x) + 1 β€ 2020 β§ (β k l, k β₯ l β§ x = β i in Finset.range (k - l + 1), 10 ^ (i + l))} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2020_a1",
"tags": [
"number_theory",
"algebra"
]
} |
Let $k$ be a nonnegative integer. Evaluate
\[
\sum_{j=0}^k 2^{k-j} \binom{k+j}{j}.
\]
| 4^k | Show that the answer is $4^k$. | null | [] | @Eq (Nat β Nat) answer fun (k_1 : Nat) =>
@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) k_1 | β β β | [
{
"t": "β",
"v": null,
"name": "k"
},
{
"t": "(β j in Finset.Icc 0 k, 2 ^ (k - j) * Nat.choose (k + j) j) = answer k",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2020_a2",
"tags": [
"algebra"
]
} |
Let $a_0 = \pi/2$, and let $a_n = \sin(a_{n-1})$ for $n \geq 1$. Determine whether
\[
\sum_{n=1}^\infty a_n^2
\]
converges. | False | The series diverges. | open Filter Topology Set | [] | @Eq Prop answer False | Prop | [
{
"t": "β β β",
"v": null,
"name": "a"
},
{
"t": "a 0 = Real.pi / 2",
"v": null,
"name": "ha0"
},
{
"t": "β n, a (n+1) = Real.sin (a n)",
"v": null,
"name": "ha"
},
{
"t": "(β L, Tendsto (fun m : β => β n in Finset.Icc 1 m, (a n)^2) atTop (π L)) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2020_a3",
"tags": [
"analysis"
]
} |
Let $a_n$ be the number of sets $S$ of positive integers for which
\[
\sum_{k \in S} F_k = n,
\]
where the Fibonacci sequence $(F_k)_{k \geq 1}$ satisfies $F_{k+2} = F_{k+1} + F_k$ and begins $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$. Find the largest integer $n$ such that $a_n = 2020$. | (Nat.fib 4040) - 1 | The answer is $n=F_{4040}-1$. | open Filter Topology Set | [] | @Eq Int answer
(@HSub.hSub Int Int Int (@instHSub Int Int.instSub)
(@Nat.cast Int instNatCastInt (Nat.fib (@OfNat.ofNat Nat (nat_lit 4040) (instOfNatNat (nat_lit 4040)))))
(@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1)))) | β€ | [
{
"t": "β€ β β",
"v": null,
"name": "a"
},
{
"t": "a = fun n : β€ => {S : Finset β | (β k β S, k > 0) β§ β k : S, Nat.fib k = n}.ncard",
"v": null,
"name": "ha"
},
{
"t": "IsGreatest {n | a n = 2020} answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2020_a5",
"tags": [
"number_theory",
"combinatorics"
]
} |
For a positive integer $N$, let $f_N$ be the function defined by
\[
f_N(x) = \sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)} \sin((2n+1)x).
\]
Determine the smallest constant $M$ such that $f_N(x) \leq M$ for all $N$ and all real $x$. | Ο/4 | The smallest constant $M$ is $\pi/4$. | open Filter Topology Set | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))) | β | [
{
"t": "β€ β (β β β)",
"v": null,
"name": "f"
},
{
"t": "f = fun N : β€ => fun x : β =>\n β n in Finset.Icc 0 N, (N + 1/2 - n)/((N + 1)*(2*n + 1)) * Real.sin ((2*n + 1)*x)",
"v": null,
"name": "hf"
},
{
"t": "answer = sSup {y | βα΅ (N > 0) (x : β), y = f N x}",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2020_a6",
"tags": [
"algebra"
]
} |
For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary (for example, $d(13) = 1+1+0+1=3)$. Let
\[
S = \sum_{k=1}^{2020} (-1)^{d(k)} k^3.
\]
Determine $S$ modulo 2020. | 1990 | The answer is $1990$. | open Filter Topology Set | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 1990) (instOfNatNat (nat_lit 1990))) | β | [
{
"t": "β β β",
"v": null,
"name": "d"
},
{
"t": "β€",
"v": null,
"name": "S"
},
{
"t": "d = fun n : β => β i : Fin (Nat.digits 2 n).length, (Nat.digits 2 n)[i]!",
"v": null,
"name": "hd"
},
{
"t": "S = β k : Icc 1 2020, ((-1 : β€)^(d k))*(k : β€)^3",
"v": null,
"name": "hS"
},
{
"t": "S % 2020 = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2020_b1",
"tags": [
"algebra"
]
} |
Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\mathbf{v} = (s_0, s_1, \cdots, s_{2n-1}, s_{2n})$ for which $s_0 = s_{2n} = 0$ and $|s_j - s_{j-1}| = 1$ for $j=1,2,\cdots,2n$. Define \[ q(\mathbf{v}) = 1 + \sum_{j=1}^{2n-1} 3^{s_j}, \] and let $M(n)$ be the average of $\frac{1}{q(\mathbf{v})}$ over all $\mathbf{v} \in V_n$. Evaluate $M(2020)$. | 1 / 4040 | Show that the answer is $\frac{1}{4040}$. | open Filter Topology Set | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@OfNat.ofNat Real (nat_lit 4040)
(@instOfNatAtLeastTwo Real (nat_lit 4040) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 4038) (instOfNatNat (nat_lit 4038))))))) | β | [
{
"t": "β β Set (β β β€)",
"v": null,
"name": "V"
},
{
"t": "β β (β β β€) β β",
"v": null,
"name": "q"
},
{
"t": "β β β",
"v": null,
"name": "M"
},
{
"t": "V = fun n β¦ ({s : β β β€ | s 0 = 0 β§ (β j β₯ 2 * n, s j = 0) β§ (β j β Icc 1 (2 * n), |s j - s (j - 1)| = 1)})",
"v": null,
"name": "hV"
},
{
"t": "q = fun n s β¦ 1 + β j in Finset.Icc 1 (2 * n - 1), 3 ^ (s j)",
"v": null,
"name": "hq"
},
{
"t": "M = fun n β¦ (β' v : V n, 1 / (q n v)) / (V n).ncard",
"v": null,
"name": "hM"
},
{
"t": "M 2020 = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2020_b4",
"tags": [
"algebra"
]
} |
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops.
Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop.
What is the smallest number of hops needed for the grasshopper to reach the point $(2021, 2021)$? | 578 | The answer is $578$. | open Filter Topology | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 578) (instOfNatNat (nat_lit 578))) | β | [
{
"t": "List (β€ Γ β€) β Prop",
"v": null,
"name": "P"
},
{
"t": "β l, P l β l.Chain' fun p q β¦ (p.1 - q.1) ^ 2 + (p.2 - q.2) ^ 2 = 25",
"v": null,
"name": "P_def"
},
{
"t": "IsLeast\n {k | β l, P ((0, 0) :: l) β§ l.getLast! = (2021, 2021) β§ l.length = k} \n answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2021_a1",
"tags": [
"geometry"
]
} |
For every positive real number $x$, let $g(x)=\lim_{r \to 0}((x+1)^{r+1}-x^{r+1})^\frac{1}{r}$. Find $\lim_{x \to \infty}\frac{g(x)}{x}$. | Real.exp 1 | Show that the limit is $e$. | open Filter Topology | [] | @Eq Real answer (Real.exp (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) | β | [
{
"t": "β β β",
"v": null,
"name": "g"
},
{
"t": "β x > 0, Tendsto (fun r : β => ((x + 1) ^ (r + 1) - x ^ (r + 1)) ^ (1 / r)) (π[>] 0) (π (g x))",
"v": null,
"name": "hg"
},
{
"t": "Tendsto (fun x : β => g x / x) atTop (π answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2021_a2",
"tags": [
"analysis"
]
} |
Determine all positive integers $N$ for which the sphere $x^2+y^2+z^2=N$ has an inscribed regular tetrahedron whose vertices have integer coordinates. | {3 * m ^ 2 | m > 0} | Show that the integers $N$ with this property are those of the form $3m^2$ for some positive integer $m$. | open Filter Topology | [] | @Eq (Set Nat) answer
(@setOf Nat fun (x : Nat) =>
@Exists Nat fun (m : Nat) =>
And (@GT.gt Nat instLTNat m (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))
(@Eq Nat
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) m
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
x)) | Set β | [
{
"t": "β",
"v": null,
"name": "N"
},
{
"t": "Set (EuclideanSpace β (Fin 3))",
"v": null,
"name": "Nsphere"
},
{
"t": "Nsphere = {p | (p 0) ^ 2 + (p 1) ^ 2 + (p 2) ^ 2 = (N : β)}",
"v": null,
"name": "hNsphere"
},
{
"t": "(EuclideanSpace β (Fin 3)) β Prop",
"v": null,
"name": "intcoords"
},
{
"t": "β p, intcoords p β β i : Fin 3, p i = round (p i)",
"v": null,
"name": "intcoords_def"
},
{
"t": "(0 < N β§ β A B C D : EuclideanSpace β (Fin 3),\n A β Nsphere β§ B β Nsphere β§ C β Nsphere β§ D β Nsphere β§\n intcoords A β§ intcoords B β§ intcoords C β§ intcoords D β§\n (β s > 0, dist A B = s β§ dist A C = s β§ dist A D = s β§ dist B C = s β§ dist B D = s β§ dist C D = s))\n β N β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2021_a3",
"tags": [
"geometry"
]
} |
Let
\[
I(R) = \iint_{x^2+y^2 \leq R^2} \left( \frac{1+2x^2}{1+x^4+6x^2y^2+y^4} - \frac{1+y^2}{2+x^4+y^4} \right)\,dx\,dy.
\]
Find
\[
\lim_{R \to \infty} I(R),
\]
or show that this limit does not exist. | $\frac{\sqrt{2}}{2} \pi \log 2$ | The limit exists and equals $\frac{\sqrt{2}}{2} \pi \log 2$. | open Filter Topology Metric | [] | @Eq Real answer
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(Real.sqrt
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
Real.pi)
(Real.log
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))) | β | [
{
"t": "β β Set (EuclideanSpace β (Fin 2))",
"v": null,
"name": "S"
},
{
"t": "S = fun R => ball (0 : EuclideanSpace β (Fin 2)) R",
"v": null,
"name": "hS"
},
{
"t": "β β β",
"v": null,
"name": "I"
},
{
"t": "I = fun R => β« p in S R,\n (1 + 2*(p 0)^2)/(1 + (p 0)^4 + 6*(p 0)^2*(p 1)^2 + (p 1)^4) - (1 + (p 1)^2)/(2 + (p 0)^4 + (p 1)^4)",
"v": null,
"name": "hI"
},
{
"t": "Tendsto I atTop (π answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2021_a4",
"tags": [
"analysis"
]
} |
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